If you posit different activities in the Godhead, then you posit different persons, per St. Gregory Nazianzus, "On Not Three Gods"
https://www.newadvent.org/fathers/2905.htm.
Creator
You cannot say that the Father is the Creator distinctly, as though this does not apply also to the Son or the Spirit, since of the Son it is written, "For by him were all things created...all things were created by him, and for him." (Col 1:16) And again, "He was in the world, and the world was made by him..." (John 1:10), and again, "He [that is, the Father] spoke [here we see the Word] and they came to be, He commanded, and they were created." (Psalm 33:9 BST)
Nor was the Spirit absent from the one act of Creation, for even in the beginning, "the Spirit of God moved upon the face of the waters" (Gen. 1:2) And again, "By his spirit he hath garnished the heavens" (Job 26:13), and, "The Spirit of God hath made me, and the breath of the Almighty hath given me life." (Job 33:4) And, "Thou sendest forth thy spirit, they are created: and thou renewest the face of the earth." (Ps. 104:30)
So the Creation is one act, and the actor is one, because the Power is one, and the energy, and the result -- all one. Therefore, the whole Godhead is the Creator: the Father is Creator, the Son is Creator, and the Spirit is Creator. You cannot say the Father alone is Creator.
I AM / YHWH
Likewise, you cannot say the Father alone is YHWH, the "I am". You yourself say that by this you mean, "existence is core to his nature. He just is." Yet if this is a property of nature, you must admit that this is shared among the Three, for it is the community of nature that indicates the unity. If you divide the nature, you divide the Godhead into three gods. But let's not only examine it logically; let's listen also to the Holy Scripture. In it, we hear the Son say of Himself, "Verily, verily, I say unto you, Before Abraham was, I am." (John 8:58) And, more than the name, the concept: "For as the Father hath life in himself; so hath he given to the Son to have life in himself." (John 5:26)
** Redeemer **
Furthermore, you cannot say of the Son, that He is the Redeemer, if by this you intend to differentiate His Person from the Father and the Spirit. For the Father is also the Redeemer, and the Holy Spirit as well. "And they remembered that God was their rock, and the high God their redeemer." (Ps. 78:35) And again, "Thou, O LORD, art our father, our redeemer..." (Is. 63:16) Neither is the redemption of Christ without the Spirit, for, "thou wast slain, and hast redeemed us to God by thy blood" (Rev. 5:9) -- that is, "the blood of Christ, who through the eternal Spirit offered himself without spot to God..." And again, "...grieve not the holy Spirit of God, whereby ye are sealed unto the day of redemption" (Eph. 4:30). So the work being one, and accomplished by the entire Godhead, "Redeemer" cannot be a differentiator amongst the Persons.
** Indweller **
Furthermore, you cannot assign "Indweller" as a unique operation or role of the Spirit alone, since the Son says, "If a man love me, he will keep my words: and my Father will love him, and we will come unto him, and make our abode with him" (John 14:23). St. Paul confirms: "That Christ may dwell in your hearts by faith..." (Eph. 3:17) St. John also confirms this word, saying (of the Father specifically): "And he that keepeth his commandments dwelleth in him, and he in him." (1 John 3:24) So then it is not only the Spirit, but the Father and the Son as well, who indwell. This cannot be a distinguishing mark, then, of only one of them.
** True Doctrine **
The true doctrine does not differentiate the Persons by their activity, which is one; nor by the results, which are one; nor by the energy, which is one; nor by the will, which is one; nor by the mind, which is one. Rather, we have learned to differentiate the Persons by their mode of existence only. Namely, that the Father is without cause, the Son is begotten of the Father, and the Spirit proceeds from the Father.
These distinctions are eternal and immutable, and without reference to Creation, since the Father is eternally the Father of the Son (else He would not be called "father"); the Son is eternally "the Son" because He is begotten of the Father; and the Spirit is the Spirit of Life from the proceeding from the Father and resting in the Son, which the Father gives to the Son "without measure". There is no time when the Son was not, and therefore there is no time when the Spirit was not proceeding from the Father to the Son.