Thank you Alex.
I am glad that you agree that the Eternal God existed prior to the birth of Mary.
It is therefore necessary to conclude that you mean something different when you quote the "Mother of God" forumlation, for it is not possible that it means that Mary pre-existed God and gave rise to this God that pre-existed her. You mean what you are saying in formulaic language and that is all I need to know.
No, it is not formulaic language if by that you mean a code word with a meaning artificially attached, like when we say "nine one one" and mean "emergency". Mother is one giving birth. Jesus Christ is God. Mary gave birth to Jesus Christ. She is mother of Jesus Christ. She is mother of God.
You mentally insert preexistence into motherhood. It is not necessarily true even with my mother, or yours, because we don't really know when my soul or yours began to exist (we know that our ensoulment predated our mothers' ensoulment, a different thing). It is surely not true with the conception and birth of Christ, because the scripture tell us so. When we ask, -- How? the scripture answers, -- Because all things are possible with God. Mary is the mother of God Who preexisted her in the most direct unconvoluted straighforward sense.