Just for the sake of clarity. Are you arguing that both the Father and the Son somehow confer personhood on the Holy Spirit?
No, not at all. The Holy Spirit proceeds from the Father and the Son, but they did not “make” him a person. He is an eternal person, just as the Father and the Son.
I think the problem people have understanding the Trinity is that they are thinking in human terms. It can be hard to grasp a concept that is so far above us–like ants trying to grasp the relationships of humans.
God simply is. Beyond all creation and before all creation, God is. He has revealed himself as three persons. In the creation story he says, “Let us create…” (Gn. 1:26) and "Behold, the man has become like one of us, knowing good and evil …{Gen. 3:22). These were internal conversations–something like one of us saying, “Come on, body, let’s get up and do” this or that. We reflect the Trinity in that we are body, mind, and spirit not merely body. We can no more say that our bodies produced our minds than we can say our spirits produced our bodies. We are one being with three dimensions.
God is more than mere human, of course. So his internal conversations are real because he really is three Persons who is one God. Jesus commanded the Apostles to go into the whole world and baptize in the name of “the Father, and the Son, and the Holy Spirit.” The Church has not based its teaching on this one verse alone, of course, but the verse does support the fact that Jesus revealed God, even more explicitly than in the OT, as one God in three persons.
The Church has explained this as three persons with one being, especially noted in our creeds. And since Jesus gave the Church the authority to decide matters of faith and morals, we accept this explanation, even if we cannot fully understand it. I certainly don’t and my explanation is not definitive, but rather limited to what you requested–a philosophical definition.