According to Trinitarianism, is God - strictly speaking - a person?

Boethius is cited by Aquinas defining person as "an individual substance of a rational nature." ST I Q29 Given this definition, since Trinitarians hold that all three persons are one Divine Substance, and since the Divine Substance is a rational nature (it is a nature with intellect), it seems fitting that God be called "a person" insofar as we refer to His nature. Aquinas also seems to think that this is fitting, saying:

"Person" signifies what is most perfect in all nature—that is, a subsistent individual of a rational nature. Hence, since everything that is perfect must be attributed to God, forasmuch as His essence contains every perfection, this name "person" is fittingly applied to God; not, however, as it is applied to creatures, but in a more excellent way; as other names also, which, while giving them to creatures, we attribute to God; as we showed above when treating of the names of God [ibid.]

Aquinas also explains the evolution of the use of person, such that it now no longer signifies substance in God, but relation subsisting in substance (or subsisting in the divine nature), but that this is only made clear after clarifications to Trinitarian doctrine are made in light of heresies.

A difficulty arises concerning the meaning of this word "person" in God, from the fact that it is predicated plurally of the Three in contrast to the nature of the names belonging to the essence; nor does it in itself refer to another, as do the words which express relation.

Hence some have thought that this word "person" of itself expresses absolutely the divine essence; as this name "God" and this word "Wise"; but that to meet heretical attack, it was ordained by conciliar decree that it was to be taken in a relative sense, and especially in the plural, or with the addition of a distinguishing adjective; as when we say, "Three persons," or, "one is the person of the Father, another of the Son," etc. Used, however, in the singular, it may be either absolute or relative. But this does not seem to be a satisfactory explanation; for, if this word "person," by force of its own signification, expresses the divine essence only, it follows that forasmuch as we speak of "three persons," so far from the heretics being silenced, they had still more reason to argue. Seeing this, others maintained that this word "person" in God signifies both the essence and the relation. Some of these said that it signifies directly the essence, and relation indirectly, forasmuch as "person" means as it were "by itself one" [per se una]; and unity belongs to the essence. And what is "by itself" implies relation indirectly; for the Father is understood to exist "by Himself," as relatively distinct from the Son. Others, however, said, on the contrary, that it signifies relation directly; and essence indirectly; forasmuch as in the definition of "person" the term nature is mentioned indirectly; and these come nearer to the truth.

To determine the question, we must consider that something may be included in the meaning of a less common term, which is not included in the more common term; as "rational" is included in the meaning of "man," and not in the meaning of "animal." So that it is one thing to ask the meaning of the word animal, and another to ask its meaning when the animal in question is man. Also, it is one thing to ask the meaning of this word "person" in general; and another to ask the meaning of "person" as applied to God. For "person" in general signifies the individual substance of a rational figure. The individual in itself is undivided, but is distinct from others. Therefore "person" in any nature signifies what is distinct in that nature: thus in human nature it signifies this flesh, these bones, and this soul, which are the individuating principles of a man, and which, though not belonging to "person" in general, nevertheless do belong to the meaning of a particular human person.

Now distinction in God is only by relation of origin, as stated above (I:28:2 and I:28:3), while relation in God is not as an accident in a subject, but is the divine essence itself; and so it is subsistent, for the divine essence subsists. Therefore, as the Godhead is God so the divine paternity is God the Father, Who is a divine person. Therefore a divine person signifies a relation as subsisting. And this is to signify relation by way of substance, and such a relation is a hypostasis subsisting in the divine nature, although in truth that which subsists in the divine nature is the divine nature itself. Thus it is true to say that the name "person" signifies relation directly, and the essence indirectly; not, however, the relation as such, but as expressed by way of a hypostasis. So likewise it signifies directly the essence, and indirectly the relation, inasmuch as the essence is the same as the hypostasis: while in God the hypostasis is expressed as distinct by the relation: and thus relation, as such, enters into the notion of the person indirectly. Thus we can say that this signification of the word "person" was not clearly perceived before it was attacked by heretics. Hence, this word "person" was used just as any other absolute term. But afterwards it was applied to express relation, as it lent itself to that signification, so that this word "person" means relation not only by use and custom, according to the first opinion, but also by force of its own proper signification. [ibid.]

So, we can say that God is "a person" if what we mean to communicate is that He is a rational nature. Just as you and I (and little babies in the womb, too) are persons owing to our rational nature, so is God, though in a way different (as must be said of every comparison between nature and God) because His nature as necessary is wholly different from natures of contingent things. ST I Q2 A3 But, by person, we mean to signify what is distinct about God, and what is distinct in Him only is relation, as His substance is supremely simple. ST I Q3

There is a complicating factor from scripture, in that the Father gives us His name "I AM" Ex 3:14 and the Son, too gives us His name "I AM" John 18:4-8. How do the Father and the Son, who are distinct persons, share a Divine Name? Is I AM a person, or no? The answer is that the Divine Persons do share personhood, by which is signified their shared Divine Essence as an intellectual substance.

It requires noting that the Summa Theologiae, the Book of Exodus, and the Gospel of John are all written in languages containing neither definite nor indefinite articles, which, it seems to me, makes this particular fact easier to express. This question of is God a person cannot even exist in Latin, Greek, or Hebrew. We must find a way to work around this in modern English, which uses such articles and thus could be confusing. Let us therefore say that, when we speak of God as we are speaking of His nature, we call Him personal, a person insofar as His nature is a personal one. But, when we speak of the members of the Trinity, let us call them persons, signifying their relations within the Divine Essence as distinct from one another.