From what I’ve read, the best definition for “person” when applied to God is “thing you can have a relationship with.” The Persons of the Trinity are distinguished by their relationships with each other. The Son is begotten of the Father; the Spirit proceeds from the Father (and/through the Son); the Father is neither begotten nor proceeding. Those are the only inherent differences between the three Persons.
Jesus, by contrast, does not have internal relationships. He has two natures, divine and human, but he is only one person, God the Son. There is not a separate human Jesus that God “possessed” or is sharing a body with. LIkewise, other people can’t have separate relationships with Jesus as a man and as God the Son. That’s why we are so insistent that Mary is properly called the bearer or mother of God – not because she is older or greater than God, as she most certainly is not, but to affirm that the son she carried and bore and brought up IS the second Person of the Trinity. You can’t divide Jesus up and say that Mary is only the mother of the human.
One popular way of expressing this is that the personhood is the “who” and the nature is the “what.” God, the Trinity, is one “what” and three “whos.” Jesus is one “who” and two “whats.”