Rather than cherry pick verses, one should begin by focusing attention on the first time God chooses to make a first person identification of Himself:

²⁶ Then God said, “Let Us make man in Our image, according to Our likeness; let them have dominion over the fish of the sea, over the birds of the air, and over the cattle, over all the earth and over every creeping thing that creeps on the earth.” ²⁷ So God created man in His own image; in the image of God He created him; male and female He created them. (Genesis 1)

Any one who begins with Genesis knows the first form of self-identification God chose was using plural masculine pronouns. Rather then deny God was confused or is unable to accurate identify Himself, the logical approach is to consider, "Who is included in us?"

Nor is this first use some mystery or grammatical aberration. The plural and singular use makes good sense. Sometimes YHVH acts or speaks singularly. That is, "He" is one of the "us" in God's first self reference.

A rational explanation for understanding why God chose to make this first reference of Himself as plural, is recognizing He is stating His nature. Attempting to deny the explicit text on the basis of the inadequacy of human definitions and terminology of persons or personhood is not an explanation for how or why God chose to first identify Himself as plural.

The plural nature of God continues throughout the Old Testament. God can be Elohim, or Eloah, or El. Why does Scripture need three different words, one of which is plural to identify God? Do translations which conflate three different words to just one, God, justify dismissing the fact God chose to first identify Himself as plural?

The Shema is often cited as "proof" of the singleness of God:

“Hear, O Israel: The LORD our God, the LORD is one! (Deuteronomy 6)
שְׁמַע יִשְׂרָאֵל יְהוָה אֱלֹהֵינוּ יְהוָה אֶחָֽד

The literal translation is Hear Israel YHVH Elohim YHVH one. Like the use of the first person pronoun, this makes perfect sense when it is understood with the passage in Genesis in view: And Elohim said, "Let us make...Hear Israel YHVH Elohim YHVH one. Moses does not say Elohim is one. He affirms YHVH Elohim and YHVH are one.

Conclusion
I believe there are two legitimate criticisms of Christian ontology of the Trinity:

1. Replacing Biblical terminology with human terms.
2. Conflating the three different Hebrew terms to a single word, God.

With respect to the first, consider this simple statement from Paul:

For through him we both have access in one Spirit to the Father. (Ephesians 2:18)
ὅτι δι᾽ αὐτοῦ ἔχομεν τὴν προσαγωγὴν οἱ ἀμφότεροι ἐν ἑνὶ πνεύματι πρὸς τὸν πατέρα

No ontology or discussions of persons, but Paul clearly states there are three each with a different role in the oneness of God:

Son: δι᾽ αὐτοῦ [through Him]
Spirit: ἐν ἑνὶ πνεύματι [in one Spirit]
Father: πρὸς τὸν πατέρα [to the Father]

Each of the three is described with a different preposition. Through, διά the Son, and with, ἐν one Spirit, and to πρὸς the Father. The last is better understood by what John later writes:

Prologue: the Word was     πρὸς τὸν θεόν  
Ephesians: the believer is πρὸς τὸν πατέρα

Paul's Trinitarian nail in the coffin is absence of the word, God. How is this oversight best explained? When Paul writes Father he means God. or when Paul writes Father he means, Father. God is clearly present in Paul's statement when the text is seen that Paul means exactly what he wrote - God is to be found through the Son, in one Spirit, πρὸς the Father.

Lastly, if translators stopped conflating three different Hebrew words as God and accurately conveyed the original text by transliterating Elohim, El, and Eloha, the reality of the triune nature of God would not be in question.

How can 3 eternal persons with 3 distinct presences, be the God of Israel, without violating the rational laws of mathematics?

Because mathematics doesn't directly enter into it, the question doesn't make sense. That is, mathematics is the science of numbers or alternatively, it is the branch of knowledge related to number as abstracted from other types of knowledge.

1person+1person+1person=1_____

There is a term which could go into that blank or rather, there is a term which conventionally would appear to fit; however, the equation suggests something which is fundamentally at odds with Trinitarian theology.

Three persons do not "make up" the 1 _____; that is, they are not "parts" or components which together constitute the entirety, which is what addition would suggest.

God is "3 Persons 1 essence or nature". (Cf, Augustine, de Trinitate VII 4,8)

Alternatively we might say "God is 3 persons of 1 essence or nature" as Aquinas does, but we must make a key distinction. In Latin, this would be expressed as "Deus est tres personas unius essentiae vel naturae"; that is, it uses what is grammatically known as the genitive case. It's important to note that it does not use any of the prepositions that might be translated as "of" such as e/ex (out of) or de (from) since the former often implies a material cause (e.g. "the table is made from wood" -> mensa est ex ligno) and the latter often implies separation, departure etc,

Initially, person was used simply to mean "what there are 3 of in God" (as Augustine put it); that is, the term was used as a name for the concept so that constant explanation would be unneeded. Incidentally, the word "person" as we use it today was not yet in common usage. Later, person was defined by Boethius to mean an individual hypostasis of a rational nature.

A being is something which has existence; more specifically, God is not only a being, but is Being. That is, God is existence and everything else that exists only exists by participating in his existence.

An essence is what makes a thing to be what it is. In particular, God because He is simple and pure act, his being (existence) is his essence -- that is, He is divinity. In contrast, an individual human has existence (being) but participates in the essence of humanity.

The Persons are distinct or distinguishable only by their relations with each other, specifically relations of opposition. The Father is not the Son because paternity is opposed to filiation (son-ship).

Personality is not a theological term; that is, there's not really a definition of personality in use that really makes sense in relation to God. Furthermore, if I say 1, you'd question why 3 persons and if I say 3, you'd say then how is God one. Perhaps the point you are aiming at with "personality" is the distinctiveness aspect, but even then, we'd have to say the only distinctiveness is the Persons and they are only distinguished by their mutually opposed relations (Father <-> Son, Father and Son <-> the Holy Spirit) since each person is the essence.

Trinitarian Ontology... What is it? Being vs person vs essence vs _______

The subject matter of the Divine Nature of the Trinity has been debated over for centuries. I have had Unitarians tell me that the term Trinity was coined in the Middle Ages. Nothing is further from the truth.

St. Augustine of Hippo (354-430) had already written 15 books on the Trinity (De Trinitate). It seems the classic works of Christendom are little read, appreciated and understood in modern times.

The Angelic Doctor St. Thomas Aquinas equally wrote on the Divine Nature of the Trinity.

The Church has many sources about it, yet many nowadays prefer to squabble over definitions already defined. People that have their minds set up on a particular outcome usually will not change their minds. Only grace can help with that. This stated I am not posting this answer for a debate.

St. Thomas Aquinas has the following to say about the Trinity and it’s Divine Nature and Essence.

Question 39. The persons in relation to the essence

1. Whether in God the essence is the same as the person?

The truth of this question is quite clear if we consider the divine simplicity. For it was shown above (I:3:3) that the divine simplicity requires that in God essence is the same as "suppositum," which in intellectual substances is nothing else than person. But a difficulty seems to arise from the fact that while the divine persons are multiplied, the essence nevertheless retains its unity. And because, as Boethius says (De Trin. i), "relation multiplies the Trinity of persons," some have thought that in God essence and person differ, forasmuch as they held the relations to be "adjacent"; considering only in the relations the idea of "reference to another," and not the relations as realities. But as it was shown above (I:28:2) in creatures relations are accidental, whereas in God they are the divine essence itself. Thence it follows that in God essence is not really distinct from person; and yet that the persons are really distinguished from each other. For person, as above stated (I:29:4, signifies relation as subsisting in the divine nature. But relation as referred to the essence does not differ therefrom really, but only in our way of thinking; while as referred to an opposite relation, it has a real distinction by virtue of that opposition. Thus there are one essence and three persons.

Article 2. Whether it must be said that the three persons are of one essence?

As above explained (I:13:2), divine things are named by our intellect, not as they really are in themselves, for in that way it knows them not; but in a way that belongs to things created. And as in the objects of the senses, whence the intellect derives its knowledge, the nature of the species is made individual by the matter, and thus the nature is as the form, and the individual is the "suppositum" of the form; so also in God the essence is taken as the form of the three persons, according to our mode of signification. Now in creatures we say that every form belongs to that whereof it is the form; as the health and beauty of a man belongs to the man. But we do not say of that which has a form, that it belongs to the form, unless some adjective qualifies the form; as when we say: "That woman is of a handsome figure," or: "This man is of perfect virtue." In like manner, as in God the persons are multiplied, and the essence is not multiplied, we speak of one essence of the three persons, and three persons of the one essence, provided that these genitives be understood as designating the form.

Article 3. Whether essential names should be predicated in the singular of the three persons?

Some essential names signify the essence after the manner of substantives; while others signify it after the manner of adjectives. Those which signify it as substantives are predicated of the three persons in the singular only, and not in the plural. Those which signify the essence as adjectives are predicated of the three persons in the plural. The reason of this is that substantives signify something by way of substance, while adjectives signify something by way of accident, which adheres to a subject. Now just as substance has existence of itself, so also it has of itself unity or multitude; wherefore the singularity or plurality of a substantive name depends upon the form signified by the name. But as accidents have their existence in a subject, so they have unity or plurality from their subject; and therefore the singularity and plurality of adjectives depends upon their "supposita." In creatures, one form does not exist in several "supposita" except by unity of order, as the form of an ordered multitude. So if the names signifying such a form are substantives, they are predicated of many in the singular, but otherwise if they adjectives. For we say that many men are a college, or an army, or a people; but we say that many men are collegians. Now in God the divine essence is signified by way of a form, as above explained (Article 2), which, indeed, is simple and supremely one, as shown above (I:3:7; I:11:4). So, names which signify the divine essence in a substantive manner are predicated of the three persons in the singular, and not in the plural. This, then, is the reason why we say that Socrates, Plato and Cicero are "three men"; whereas we do not say the Father, Son and Holy Ghost are "three Gods," but "one God"; forasmuch as in the three "supposita" of human nature there are three humanities, whereas in the three divine Persons there is but one divine essence. On the other hand, the names which signify essence in an adjectival manner are predicated of the three persons plurally, by reason of the plurality of "supposita." For we say there are three "existent" or three "wise" beings, or three "eternal," "uncreated," and "immense" beings, if these terms are understood in an adjectival sense. But if taken in a substantive sense, we say "one uncreated, immense, eternal being," as Athanasius declares.

Article 4. Whether the concrete essential names can stand for the person?

Some have said that this name "God" and the like, properly according to their nature, stand for the essence, but by reason of some notional adjunct are made to stand for the Person. This opinion apparently arose from considering the divine simplicity, which requires that in God, He "who possesses" and "what is possessed" be the same. So He who possesses Godhead, which is signified by the name God, is the same as Godhead. But when we consider the proper way of expressing ourselves, the mode of signification must be considered no less than the thing signified. Hence as this word "God" signifies the divine essence as in Him Who possesses it, just as the name "man" signifies humanity in a subject, others more truly have said that this word "God," from its mode of signification, can, in its proper sense, stand for person, as does the word "man." So this word "God" sometimes stands for the essence, as when we say "God creates"; because this predicate is attributed to the subject by reason of the form signified—that is, Godhead. But sometimes it stands for the person, either for only one, as when we say, "God begets," or for two, as when we say, "God spirates"; or for three, as when it is said: "To the King of ages, immortal, invisible, the only God," etc. (1 Timothy 1:17).

Article 5. Whether abstract essential names can stand for the person?

Concerning this, the abbot Joachim erred in asserting that as we can say "God begot God," so we can say "Essence begot essence": considering that, by reason of the divine simplicity God is nothing else but the divine essence. In this he was wrong, because if we wish to express ourselves correctly, we must take into account not only the thing which is signified, but also the mode of its signification as above stated (Article 4). Now although "God" is really the same as "Godhead," nevertheless the mode of signification is not in each case the same. For since this word "God" signifies the divine essence in Him that possesses it, from its mode of signification it can of its own nature stand for person. Thus the things which properly belong to the persons, can be predicated of this word, "God," as, for instance, we can say "God is begotten" or is "Begetter," as above explained (Article 4). The word "essence," however, in its mode of signification, cannot stand for Person, because it signifies the essence as an abstract form. Consequently, what properly belongs to the persons whereby they are distinguished from each other, cannot be attributed to the essence. For that would imply distinction in the divine essence, in the same way as there exists distinction in the "supposita."

Article 6. Whether the persons can be predicated of the essential terms?

As above explained (Article 5), although adjectival terms, whether personal or notional, cannot be predicated of the essence, nevertheless substantive terms can be so predicated, owing to the real identity of essence and person. The divine essence is not only really the same as one person, but it is really the same as the three persons. Whence, one person, and two, and three, can be predicated of the essence as if we were to say, "The essence is the Father, and the Son, and the Holy Ghost." And because this word "God" can of itself stand for the essence, as above explained (Article 4, Reply to Objection 3), hence, as it is true to say, "The essence is the three persons"; so likewise it is true to say, "God is the three persons."

Article 7. Whether the essential names should be appropriated to the persons?

For the manifestation of our faith it is fitting that the essential attributes should be appropriated to the persons. For although the trinity of persons cannot be proved by demonstration, as was above expounded (I:32:1, nevertheless it is fitting that it be declared by things which are more known to us. Now the essential attributes of God are more clear to us from the standpoint of reason than the personal properties; because we can derive certain knowledge of the essential attributes from creatures which are sources of knowledge to us, such as we cannot obtain regarding the personal properties, as was above explained (I:32:1. As, therefore, we make use of the likeness of the trace or image found in creatures for the manifestation of the divine persons, so also in the same manner do we make use of the essential attributes. And such a manifestation of the divine persons by the use of the essential attributes is called "appropriation."

Mystery

The very concept of a "Trinity" God is shrouded in ambiguity to the point most Trinitarians agree that it is an unexplainable mystery--both to themselves and to everyone else. This is why the expression "mystery of the Holy Trinity" is so common.

The fact is, there are multiple strains of Trinitarian doctrine, and not all Trinitarians will agree on the most basic definitions of the words person, personality, being, or essence.

For example, Catholics teach, and always have, that God is one being: but Seventh-day Adventist Trinitarians presently teach that God is three beings, though their Trinitarian doctrine began with God as a single being, prior to which the Seventh-day Adventists did not accept the Trinity at all.

Variety of Views

A careful explanation of some of the nuances within Adventist Trinitarianism, and indeed, inclusive of most Protestants, can be found HERE, of which I will quote a representative portion below:

There are two integral factors to a trinity doctrine. This is ‘threeness’ and ‘oneness”. As Paul Petersen wrote (as Chair of Religion & Professor of the Hebrew Bible, Andrews University):

“The core elements of the doctrine of the Trinity are oneness and distinctiveness. The Father, the Son, and the Holy Spirit are one, yet three. To express this conviction, words and expressions came into use that are not explicitly used in the Bible. The oneness of God we confess by claiming that God is one in being; the distinctiveness we confess by teaching that there are three persons.” (Paul Petersen, Andrews University, May 2015, page 3, ‘God in 3 Persons — in the New Testament’)

While all Trinitarians agree on this core concept, within Trinitarianism there can be notable variation from one denomination to another or even variants within each denomination. That said, there too exists within Adventism, two main notable variations, namely the tritheistic and the orthodox variety. . . .

In general, there are three main categories of how trinity or triunity is understood:

1. Consubstantial trinity – This version of trinity is characterized as a single divine “Being,” comprised of three “Persons”, sharing one indivisible (undivided) substance (Catholic/Orthodox trinity and many theologian class of SDA Church who are heavily influenced by evangelical authors would fall under this category ). (“One Being” = “3 Persons”; “Being” and “Person” is distinguished)

2. Modalistic trinity – This version of trinity is comprised of three modes or roles occupied by single divine individual wherein one God is manifesting or revealing himself in three different ways, without distinct and coexisting persons in the divine nature. (i.e. Father is Jesus, Father is the Holy Spirit, etc.; same person operating in 3 different modes)

3. Tritheistic trinity – The “trinity” is comprised of three distinct divine “persons” but they understand “persons” as individual Beings; who all have same power, same nature, purpose, etc. Because they are in agreement in everything they do, they are said to be “one God,” not that all three make up a single Being (Majority of Seventh-day Adventist trinitarians would fall under this category).

“There were some very early that turned the doctrine of the Trinity into Tritheism, and, instead of three divine persons under the economy of Father, Son, and Holy Spirit, brought in three collateral, co-ordinate, and self-originated beings, making them three absolute and independent principles, without any relation of Father or Son, which is the most proper notion of three gods. And having made this change in the doctrine of the Trinity, they made another change answerable to it in the form of baptism.”–Antiquities, b. 11, chap. 3, § 4.

J. H. Waggoner, Signs of the Times, January 27, 1881, pg. 40

https://documents.adventistarchives.org/Periodicals/ST/ST18810127-V07-04.pdf

In general, SDA Trinitarians have made the Holy Spirit into a third “being” that is essentially the same as the Father and Son, to the point where any of them could just as well swap roles and it wouldn’t make any difference. The Father could have taken the role of the Son, the Son could have taken the role of the Spirit, the Spirit could have taken the role of the Father, etc (you can find statements by SDA theologians surmising to these sort of ideas further down in this article). This is different than the Trinity doctrine taught by other churches, which firmly declares that the Father, Son, and Holy Spirit are three “persons” inside a SINGLE “being”. So Sunday-keeping Trinitarians (as well as Catholics) teach three persons that make up one being, whereas Adventist teach three “beings” that are likened to 3 individual divine beings belonging to a single family.

Changing Definitions

One of the biggest problems with understanding the basic elements of Trinitarian belief is that the very word definitions we use have not always carried the same meaning: their definitions have evolved over time.

Person or Being?

The word "person" today usually implies "being." This is at the core of the issue with the doctrinal shift in some denominations toward tritheism (three God beings). While most of these Tritheists still consider themselves to be Monotheists, by strict dictionary definitions, they are not.

The word "person" had nothing to do with a "being" when it first came into use. A review of the dictionary definition for this word, as printed in Noah Webster's 1828 Dictionary, will suffice to show this.

PERSON, noun per'sn. [Latin persona; said to be compounded of per, through or by, and sonus, sound; a Latin word signifying primarily a mask used by actors on the stage.]

1. An individual human being consisting of body and soul. We apply the word to living beings only, possessed of a rational nature; the body when dead is not called a person. It is applied alike to a man, woman or child.

A person is a thinking intelligent being.

2. A man, woman or child, considered as opposed to things, or distinct from them.

A zeal for persons is far more easy to be perverted, than a zeal for things.

3. A human being, considered with respect to the living body or corporeal existence only. The form of her person is elegant.

You'll find her person difficult to gain.

The rebels maintained the fight for a small time, and for their persons showed no want of courage.

4. A human being, indefinitely; one; a man. Let a person's attainments be never so great, he should remember he is frail and imperfect.

5. A human being represented in dialogue, fiction, or on the stage; character. A player appears in the person of king Lear.

These tables, Cicero pronounced under the person of Crassus, were of more use and authority than all the books of the philosophers.

6. Character of office.

How different is the same man from himself, as he sustains the person of a magistrate and that of a friend.

7. In grammar, the nominative to a verb; the agent that performs or the patient that suffers any thing affirmed by a verb; as, I write; he is smitten; she is beloved; the rain descends in torrents. I, thou or you, he, she or it, are called the first, second and third persons. Hence we apply the word person to the termination or modified form of the verb used in connection with the persons; as the first or the third person of the verb; the verb is in the second person.

8. In law, an artificial person is a corporation or body politic.

In person by one's self; with bodily presence; not be representative.

The king in person visits all around.

PER'SON, verb transitive To represent as a person; to make to resemble; to image. [Not in use.]

So a person was actually more equivalent to a role; especially as could be applied to God, who is clearly not a human. Note that of the eight senses of meaning provided by Webster, only the sixth could reasonably be applied to God, as God is neither human, nor a mere point of grammar. Clearly this "person" does not connote "being."

But modern dictionaries have changed the rhetoric. For example, Wikipedia defines person as:

A person (PL: people) is a being who has certain capacities or attributes such as reason, morality, consciousness or self-consciousness, and being a part of a culturally established form of social relations such as kinship, ownership of property, or legal responsibility. The defining features of personhood and, consequently, what makes a person count as a person, differ widely among cultures and contexts.

[NOTE: Wikipedia's references clashed with those of Stack Exchange, and have been stripped.]

What was once a role or character of office has morphed into a being.

But Wikipedia also notes the historical development of the word:

Development of the concept

In ancient Rome, the word persona (Latin) or prosopon (πρόσωπον; Ancient Greek) originally referred to the masks worn by actors on stage. The various masks represented the various "personae" in the stage play.

Those who are oblivious to these historical changes are the most susceptible to accepting revised beliefs that were not accepted in previous times.

Summary

A person is a role, and a single being can have more than one role; therefore a single being can be represented as multiple persons. As an example, the same man might relate to his family members in the person of a father, or that of a son, or that of a husband. A multi-being God, however, implies Tritheism, and technically no longer represents monotheistic belief--or the belief in "one God."

The word "essence" is, to the author of this post, beyond discussion with respect to God. No human can know what God is made of, and His essence, or substance, is not addressed in Scripture, nor should it be mentioned. However, many do presume to address it--I will let them find their own manner of defending such.

The question your asking is flawed. "According to the triune concept, there exist 3 eternal persons." That is "NOT" the concept of the Trinity. Let me now define "being" and "person."

Being: the state or fact of existing; existence. something that exists or is thought to exist.

Person/Personality: is the combination of characteristics or qualities that form an individuals distinctive character.

Now, by "Trinity" I mean within the nature of the ONE true God, there are three eternal, distinct Persons: the Father, the Son, and the Holy Spirit. These three persons are the ONE God. By using the logical technique called the transitive property of equality (things equal to the same thing are equal to each other).

The Bible teaches that there is only one uncreated, eternal, true God (Exodus 3:14; Deuteronomy 6:4; John 17:3). The Bible teaches that there will never be any other uncreated eternal, true God(s) to come into existence (Isaiah 43:24; 44:6-8; 1 Corinthians 8:4-6; Galatians 4:8).

The Bible teaches that this ONE true God created everything ex-nihilo, or from nothing (Psalm 90:2; Isaiah 44:24; Hebrews 3:4; 11:3).

When we say "the Father is God," we mean that a distinct person known as the Father is identifiable as the ONE true God, the creator ex-nihilo (2 Peter 1:17). When we say "the Son is God," we mean that a distinct person known as the Son is identifiable as the ONE true God, the creator ex-nihilo (John 1:1-3; Colossians 1:16).

When we say "the Holy Spirit is God," we mean that a distinct person known as the Holy Spirit is identifiable as the ONE true God, the creator ex-nihilo (Job 33:4; Acts 5:3-4; 1 Corinthians 2:11).

Therefore, as the transitive property of equality shows us, the three Persons Are the One God, (Matthew 28:19). It has been asked that there are so many scriptures that only use Lord, God and many other titles referring to God. God said my glory will I not give to another...Is this all 3 speaking or is it only one of the three persons?

Where no distinction is made, it is obviously unnecessary to make a distinction. The persons of the Trinity are persons in relation to each other, any one of the persons in relation to us is simply God. If you are in a relationship with the Father/Son/Holy Spirit, then you must be in a relationship with them all, for there is only ONE God. If you deny one, you deny all.