According to Trinitarians, who is the begotten "son" in Psalm 2 and what happened on this "day"?

The Athanasian creed states:

...The Son is of the Father alone; not made, nor created; but begotten. ... And in this Trinity none is before, or after another; none is greater, or less than another. But the whole three Persons are coeternal, and coequal.

The Nicene Creed as used in the Roman Catholic mass, states:

Credo ... in unum Dóminum, Iesum Christum, Fílium Dei unigénitum, et ex Patre natum ante ómnia sǽcula. Deum de Deo, lumen de lúmine, Deum verum de Deo vero, génitum, non factum, consubstantiálem Patri: per quem ómnia facta sunt.

So, in Catholic (very trinitarian indeed) faith, the answers to your questions can be as follows:

1. Before all time. Not a "day", because all was created "per" (through) Him. So logic dictates that there was no day, no time, when He was "genitum" (begotten)
2. The Son of the Father, second Person of the Trinity, is Jesus Christ indeed.
3. This is not answered by the creeds I used to answer your questions I think, so I leave this one to someone else

OP Begotten - to be brought into existence by the seed of a parent

Question(s) - answer as many as you are willing and able:

What day was this "son" begotten? Can we assume this "son" is Jesus? Was this the "day" YHWH also enthroned him in v16?

While we know that Christ has always existed, that there was never a time when He did not exist, this verse is about the Son of God as to His humanity (and Godship). So, let's use the OP understanding of "begotten" in this sense.

Therefore the Lord himself shall give you a sign; Behold, a virgin shall conceive, and bear H3205 a son, and shall call his name Immanuel. Isa 7:14

The word "bear" is the same as "begotten". It means the duration from conception to birth. See Gesenius.

This definition is shown also at Heb 1:5-6, being this day

For unto which of the angels said he at any time, Thou art my Son, this day have I begotten thee? And again, I will be to him a Father, and he shall be to me a Son? And again, when he bringeth in the first begotten into the world, he saith, And let all the angels of God worship him.

What day was this "son" begotten?

While many believe the idea Christ was born 12/25 and some believe a time in January, in keeping with biblical time frames, however, He was most likely born about the Fall equinox on the first day Feast of Booths (Sept-Oct). His conception 9 months earlier would be about the winter solstice (Dec). Recall John the Baptist, He must increase (Jn 3:30).

Having said that, it is also clear that at the resurrection this same declaration is made.

God hath fulfilled the same unto us their children, in that he hath raised up Jesus again; as it is also written in the second psalm, Thou art my Son, this day have I begotten thee. Acts 13:33

Further, there is a sense of begotten as Son at His annointing (baptism) by John the Baptist.

So also Christ glorified not himself to be made an high priest; but he that said unto him, Thou art my Son, to day have I begotten thee. Heb 5:5

That is in reference to His baptism by John in Matthew 3:13-17.

In short, there are three references to "begotten" that apply to Jesus Christ from His conception and birth, to His baptism, to His resurrection.

OP Can we assume this "son" is Jesus?

The word "anointed" means Messiah. It may mean a prophet, king, or priest. Christ fulfilled all three offices. The prophetic was His conception and birth. The priestly was at His baptism. The kingly at His resurrection.

OP Is this shown in Psalm 2, is it prophetic or fulfilled in David's time?

Clearly it was not fulfilled in David. God, however, had made a promise about David's seed (and Abraham's seed).

And when thy [David] days be fulfilled, and thou shalt sleep with thy fathers, I will set up thy seed after thee, which shall proceed out of thy bowels, and I will establish his kingdom. 2 Sam 7:12

So, yes, this Psalm refers to Christ.

Was this the "day" YHWH also enthroned him in v16?

Not sure to what v16 refers, but v6 says this.

Yet have I set my king upon my holy hill of Zion.

When?

So then after the Lord had spoken unto them, he was received up into heaven, and sat on the right hand of God. Mar 16:19

This was upon His ascension on day 40 after resurrection (inclusive counting). 10 days later He sent the Spirit.

So, the day the Son was begotten is tri-fold (conception to birth, baptism, resurrection). It does refer to Jesus Christ (the promised seed). He was enthroned on day 40 following His resurrection.

EDIT TO ADD:

The "begotten" definition used by the OP is not a problem, keeping in mind the nature of the Son of God.

OP Can YHWH beget Himself? Do fathers ever beget themselves? (If this son is Jesus and Jesus is YHWH)

No, a body was prepared by the Father (Heb 10:5 in reference to Septuagint Psalm 40:6-7).

Did Jesus beget a son? (If this son is NOT Jesus and Jesus is YHWH).

Spiritually He has begotten the sons of God.

For ye are all the children of God by faith in Christ Jesus. Gal 3:26

You must be born again (John 3:7)

I will declare the decree: YHWH hath said unto me, Thou art my Son; this day have I begotten thee
אֲסַפְּרָה אֶֽל חֹק יְֽהוָה אָמַר אֵלַי בְּנִי אַתָּה אֲנִי הַיּוֹם יְלִדְתִּֽיךָ

This day is how הַיּוֹם is rendered, the day is the correct meaning. Suppose this passage was understood as referring to David. On which day was he begotten? The day he was conceived? The day he was born? The day he was anointed as king by Samuel? The day he was anointed as king by the men of Judah? The day he was anointed as king by the elders of Israel?

The verse is prefaced by I have set My king on My holy hill of Zion. We might assume the day was the day David was anointed as king by the elders of Israel, since that was the day David was officially recognized as king over all Israel. On the other hand, I have set My king... is God speaking and He would hardly need to wait for the elders of Israel to finally agree and confirm His decision.

The point is the day can never be limited to a single day in the life of David and it is unreasonable to demand an understanding with respect to Jesus which is impossible to apply to David. It took three separate days before David was fully recognized as king of Israel; three different occasions on which he was anointed as king. Therefore if David is the example, Jesus would have three days: birth, death, resurrection.

The Biblical Hebrew of portion of Psalm 2 in question is quite interesting:

YHWH hath said unto me, Thou art my Son
יהוה אמר אלי בני

...YHWH said to me My son... Yet, elsewhere אלי is translated as my God:

My God my God why have you forsaken me
אלי אלי למה עזבתני

As a Trinitarian I prefer to treat אלי in Psalm 2 exactly as in Psalm 22:

I will declare the decree: YHWH hath said My God My Son, the day have I begotten thee

When אלי is understood as My God the specific day in question is of little importance. However, I would note the letter to the Hebrews has a similar understanding of God calling the Son, God:

But of the Son he says, “Your throne, O God, is forever and ever, the scepter of uprightness is the scepter of your kingdom. (Hebrews 1:8 ESV)

Therefore, the day is the one of exaltation. when Jesus returns to His Father who says to Him, My God My Son.

According to Trinitarians, who is the begotten "son" in Psalm 2 and what happened on this "day"?

As a trinitarian myself I believe, along with others, the "day" ("This day have I begotten you") is the Day of Christ's Resurrection from the dead when his dead human body was raised to life and given an immortal, resurrection body to reign forever as the God-man, in accordance with Acts 13:33.

"This day have I begotten thee" does not refer to a day at the beginning of eternity nor to his being "eternally begotten".

There are (at least) three quotes of Psalm 2:7 in the NT:-

In Acts 13:33, the emphasis is on the words "this day". Acts 13:28-34 reads:

28 And though they found in him no guilt worthy of death, they asked Pilate to have him executed. 29 And when they had carried out all that was written of him, they took him down from the tree and laid him in a tomb. 30 But God raised him from the dead, 31 and for many days he appeared to those who had come up with him from Galilee to Jerusalem, who are now his witnesses to the people. 32 And we bring you the good news that what God promised to the fathers, 33 this he has fulfilled to us their children by raising Jesus, as also it is written in the second Psalm, “‘You are my Son, today I have begotten you.’ 34 And as for the fact that he raised him from the dead, no more to return to corruption, he has spoken in this way, “‘I will give you the holy and sure blessings of David.’ (ESV)

Now we have a clearer idea from Acts 13:28-34 of what is being referred to by the word "begotten" in Psalm 2:7, we can get yet more detail from the phrase in Revelation 1:5,

"the firstborn of the dead".

This shows that "begotten" in Psalm 2:7 is referring to Christ's being raised to life from the dead. Rev 1:5 confirms the resurrection of our Lord is being referred to. He is the first to be raised from the dead with the immortal, resurrection body.

(By the way, notice the way the NIV uses dynamic equivalence and I think strays too far from the Greek in the following verses Acts 13:33, Hebrews 1:5, Hebrews 5:5, giving the wrong sense. "Today I have become your Father" leads away from the idea of "Today I have begotten you", i.e. risen you to new life (from the dead).)

In Hebrews 1:5 the emphasis is on the word "son". And who is this "son"? Hebrews 1:5-8 says:

5 For to which of the angels did God ever say, “You are my Son, today I have begotten you”? Or again, “I will be to him a father, and he shall be to me a son”? 6 And again, when he brings the firstborn into the world, he says, “Let all God's angels worship him.” 7 Of the angels he says, “He makes his angels winds, and his ministers a flame of fire.” 8 But of the Son he says, “Your throne, O God, is forever and ever, the scepter of uprightness is the scepter of your kingdom. (ESV)

And in Hebrews 5:5, the emphasis is that the man Jesus Christ did not choose the honour of being the Son of God, but God chose him. This choosing is in the context of our Lord Jesus becoming the High Priest: the man Jesus Christ did not choose to be the eternal High Priest, but it was God's choice:

4 And no one takes this honour for himself, but only when called by God, just as Aaron was. 5 So also Christ did not exalt himself to be made a high priest, but was appointed by him who said to him, “You are my Son, today I have begotten you” (Heb 5:4-5, ESV).

So the word "son" refers to our Lord Jesus Christ who is so far superior to the angels that they must worship him and, according to Hebrews 1:8, "God".

To be sure to answer all your other questions directly:

What day was this "son" begotten?

He was begotten on the day of his resurrection.

Can we assume this "son" is Jesus?

No assumptions are needed: it is clear from Hebrews ch1, ch5 and Acts 13-28-34 that Psalm 2 refers to our Lord Jesus Christ.

Was this the "day" YHWH also enthroned him in v16?

Uuh? Verse 16 of what passage?

Can YHWH beget Himself? Do fathers ever beget themselves? (If this son is Jesus and Jesus is YHWH)

YHWH is a term for the triune God. Sometimes in scripture the term refers to the Father, sometimes to the Holy Spirit, sometimes to Jesus Christ the Son, and sometimes to all the persons in the Godhead.

I believe the idea of begetting is not very useful when referring to the divine nature of Christ because it suggests there was a time when Christ did not exist, and saying he is "eternally begotten" is a contradiction in terms. Quoting Dr Robert Reymond in "A New Systematic Theology of the Christian Faith", page 329,

He [John Calvin] concludes his treatment of the doctrine of the Trinity by declaring the ancient doctrine of "eternal generation" to be of "little profit", unnecessarily "burdensome", "useless trouble", and "foolish": 'While I [says Calvin] am zealous for the edification of the church, I felt that I would be better advised not to touch upon many things that would profit but little, and would burden my readers with useless trouble. For what is the point in disputing whether the Father always begets? Indeed, it is foolish to imagine a continuous act of begetting, since it is clear that three persons have subsisted in God from eternity.' (Calvin's Institutes book I, chapter xiii, 29).

The New International Version chooses "the one and only" in places where the KJV uses "only begotten". The idea in John 3:16 and other places in the NT is that of uniqueness. Jesus is "the one and only Son of God", in a way that no other is or ever can be. The Scriptures are clear, "God was manifest in the flesh", 1 Tim 3:16. Those who use the term "begotten" in John 3:16 and the like passages want to emphasise that Jesus is of the same nature as the Father, i.e. that Jesus is God just as much as the Father is God. But this idea of equality is clear from Jesus calling God his Father (see John 5:18).. every time he called God his Father he was claiming to have the same nature as God the Father. Introducing the idea of the Son of God being "eternally begotten" to prove his deity is foreign to the NT.

In Psalm 2 the emphasis seems to be on the promotion of Christ's human nature to an immortal body in order to reign forever in glory, i.e. the receipt of a resurrection body. Now a human is reigning in glory, as the God-man.

Did Jesus beget a son? (If this son is NOT Jesus and Jesus is YHWH).

No. Isaiah 53:8(?).

Since posting this answer, the OP has changed the question so significantly that this answer will not seem to fit in with the edited question. Sticking to the OPs initial requirement to answer according to his definition (as in the next paragraph), the answer makes sense.

The definition you use, "Begotten - to be brought into existence by the seed of a parent", is not the only biblical definition of the word in question. If you do not consider the other definition (which is the one in the Athanasian Creed) you will make no sense of the Athanasian Creed which equates with the sense all Trinitarians have. This means that the only answers Trinitarians can give to this question about Psalm 2:6-7, if sticking to only your definition, are limited to the Old Testament understanding, as follows:

1. What 'day' was this son begotten? - This son is ascribed to being king David, the 'Anointed One' of verse 2. See 1 Samuel 16:1-13. Of course, God did not literally father David! Everybody knows that David's father was Jesse. This, in itself, proves that the literal definition insisted upon by the OP is inadequate. But, sticking to that, the 'day' in question could be the day Samuel anointed David as king-to-be. Or it could be the day David ascended the throne, after Saul's death. Either could be the literal 'day' when David became God's Anointed One. Here is what David himself said - his last words:

"The oracle of David son of Jesse, the oracle of the man exalted by the Most High, the man anointed by the God of Jacob, Israel's singer of sings: The Spirit of the Lord spoke through me; his word was on my tongue. The God of Israel spoke, the Rock of Israel said to me..." 2 Samuel 23:1-7.

2. Can we assume this son is Jesus? - No, not if we remain restricted to the definition of the OP. The only way anybody can start to consider whether Jesus Christ, the Son of God, is prophetically implied in Psalm 2, is to include what New Testament writers state. It's significant, however, that they state the initial application is to king David, before moving on to claim that Jesus Christ is the fulfillment of the prophetic aspect of Psalm 2. In Acts 2:24-36 the apostle Peter unpacks this massively important interpretation. But, if answers must stick to the definition of the OP, then it goes beyond the limited scope of the question to quote those verses, which actually show that Jesus Christ, the Son of God, is the fulfilment of the prophecy about God's Anointed One.

3. Was this the day YHWH also enthroned him in verse 16? - In Acts 2:31-36 king David is quoted as saying, "The Lord said to my Lord, 'Sit at my right hand until I make your enemies a footstool for your feet'." Then the application is made to Jesus Christ being resurrected from the dead and ascending to heaven, to sit at God's right hand. However, given the limited definition allowed for answers, the only answer as to when God enthroned king David is what I mentioned in No. 1. But as there is no verse 16 in Psalm 2, I can only assume you mean verses 4 to 6 which speak of:

"The One enthroned in heaven laughs..." and who then says in verse 6 that he has installed his King on Zion. That is God speaking. God is this One enthroned in heaven, prior to installing his King on Zion. Of course, the day God enthroned king David on Zion was not the same day David prophetically pointed to when the Son of God was resurrected to sit at the right hand of God in heaven.

This is, to Trinitarians, an unsatisfactory answer, because we are not being 'allowed' to apply the other definition of 'begotten' to the matter. Whenever Jesus Christ is spoken of as the 'only-begotten' Son of God, it has nothing to do with physical, literal, sexual fathering of a child. Really, a fresh question needs to be asked, 'allowing' for this other biblical meaning of 'only-begotten', which speaks of a unique relationship between God the Father and God the Son, an eternal relationship in the Godhead, as the Athanasian Creed speaks of.