Michael, there is nothing you have written that indicates God as a compound being consisting of three divine Persons.
When Jesus said, “The Father and I are one,” He referred to the fact that He and the Father are in one in thought, in purpose, etc. Also that HE is the exact imprint of the Father’s essence, in other words another divine Individual who is exactly like the Father, but yet a different Individual. Suppose I take a picture of myself out of my left pocket and show it to you. Then I say, “I am going to show you another picture of me,” and I bring out from my right pocket another picture of me made from the same negative. You might exclaim, “But that’s the same picture!” Then I might say, "No, it’s not. And holding up the first, I might say, “Here is one picture,” and then holding up the other, “And here is the other picture. Two pictures!” The Father and the Son are like that. They are two Individuals, but are exactly alike, so that Jesus was able to say, “If you’ve seen me, you’ve seen the Father.”
As for John 1:1, this sentence is not affirming that the Logos (the Son of God) was both with God and was God. That is illogical—and I mean truly illogical, not merely unreasonable. It would be analogous to a boy saying to you, “My father is with me today. In fact I AM my father.” The construction of John 1:1 indicates that the first occurrence of the word “God” refers to the Father. For it has the article. The second occurrence of the word “God,” in the same verse, does NOT have the article. If the sentence were in natural order then the New World Translation of Jehovah’s Witnesses would be correct, “And the word was a god.” But the order of the words is, “God was the Word (Logos).” This means that “God” or “Deity” is the kind of thing (the essence) that the Word was. For our modern day it might better be translated,“And the Word was divine” or perhaps more accurately, "The Word was divine Essence."This was a way of saying that the Word, the Son of God, was of the same divine essence as God. That was because God begat his Son as his first act, and then using his Son as his Agent, created the Universe THROUGH Him.
Second-century Christian writers affirmed that the begetting of the Son was the first act of God, and occurred "before all ages."Indeed, even in the fourth century, the original Nicene Creed stated that Christ was “begotten before all ages,” and the early Trinitarians of that period accepted the statement. But later Trinitarians saw the statement as inconsistent with Trinitarianism, and so they changed it to “eternally begotten” (whatever that means).
As for the Holy Spirit, Paul affirms that the Lord [Jesus] IS the Spirit (2 Cor 3:15). As I see it, the Persons of the Father and the Son are not confined to any place, but They can extend their Persons to any place in the Universe, and that is their Spirit. That Spirit is ONE because the Persons of the Father and Son are fully united. The phrase “spirit of God” occurs 12 times in the New Testament, and the phrase “spirit of Jesus” occurs twice, “spirit of Christ” twice, “spirit of Jesus Christ” once, and “spirit of his Son” once.
In one sense there is only one God, that is, the Father. In a prayer to God, Jesus said:
And this is eternal life, that they know you the only true God, and Jesus Christ whom you have sent. (John 17:3)
In saying this, Jesus distinguished Himself from “the only true God.”
But in the sense of being divine, Jesus may also be said to be “God” but not in the same sense that the Father is the one true God. However, Jesus and his Father are not joined together into one compound God (Binitarianism); they are two distinct divine Individuals, but are of the same divine essence. You and your son are two distinct individuals but are of the same human essence. (However two human beings can never be exactly the same as is the Father and the Son—not even “identical twins).”