You are thinking too much like a human. A God capable of anything can defy our logic, no?
Who is to say that God cannot divide Himself three ways? Jesus seems more Human to us, we can relate to Him because He came to Earth to relate to us. You know about Moses...He saw only PART of God and his glory, as God was going away from him, and his HAIR TURNED WHITE! If God came as the Father, we would all surely die from His magnificence. That may be a reason that God the Father stayed in Heaven. Also, God (Jesus) could not experience what it is like to depend on God (the Father) unless the Father was in Heaven to depend on. Now, the Holy spirit may have been somewhere on Earth, or all around even. The significance of that Pentacost was that it was the first time that the Holy Spirit (the spirit of God) entered INTO His people. So, maybe the Holy Ghost WAS on Earth with Jesus, it just wasn't in His people until later.
It is confusing and against Human logic, but remember that God never promised He would make sense to us. He just told it like it is.
As to the interpretation of that passage, while you're analysis hold's some merit, I don't think it tells the whole story. Jesus is saying to trust in Him. If God has only one personage, why would he say: "trust in Me, and trust also in Me?"
Another question: Why did Jesus tell us to baptise in the name of the Father, the Son, and the Holy Spirit?
God bless and keep you,
Amadeus