This discussion all seems weird to me, because this is elementary church history or doctrine. I’m going to give you traditional Catholic definitions. The East may not precisely agree, but it’s similar. I'm taking this from the section on the Trinity in the Summa, from newadvent.org.
If you want a clear definition, Aquinas defines a person as “a subsistent individual of a rational nature.” Hypostasis is more general, as it is not necessarily rational. However since God is rational, in the context of the Trinity hypostasis and person are the same. (Aquinas also notes differences between God and humans, so that you have to be careful about the exact sense of individual and rational when referring to God.)
With normal people, it’s obvious that we are subsistent individuals, and one hopes most of us are rational.
However the question is whether God is three subsistent individuals. That kind of sounds like tritheism, the criticism Hoghead1 was making.
Aquinas, however, wants to use the classical definition of person, but he also wants to say that in the Trinity the persons are defined only by their relations. So the arguments is this: persons are by definition individuals. But what it means to be individual depends upon your nature. For humans it means having separate flesh, bones and soul, because that’s the nature of humans.
But, he argues, distinction in God is only by relation. In particular, the Father is paternity, the Son is begotten from him and the Holy Spirit proceeds from him. [I’m extrapolating from what he actually said.] Therefore for the divine persons, distinction and individuality come from their relations to each other, the Father being source, etc.
He connects this with traditional substance metaphysics by saying that “… 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 while person / hypostasis and relation are often used as synonyms, in fact person “signifies relation not as such, but by way of a substance--which is a hypostasis.” My dummy version of this is that you can’t have relations in the abstract. It has to be between two things. But in this case the things involved are distinguished only by their relations with each other.
At least this is what I get from him, given that some of his metaphysical statements don’t make much sense to me.
My comment would be that this seems like the best one can do given the starting point of traditional definitions. But I’m not convinced that bringing substance into it helps much. I’d prefer simply to say that God is personal and has the experience of personal relationship, but I don’t think I’d try to go further in defining it metaphysically.
However the Trinity exists not just as a way of saying that God as personal, but as a consequence of a specific history of Christological arguments. But I'm even less convinced that substance metaphysics is the right way to deal with the Incarnation, even though Aquinas manages to interpret it in a way that deals with some of the potential shortcomings.