The term “God” in the title “Mother of God” has ALWAYS meant “the Second Person of the Trinity.”
ALL of the “ambiguity” you write about is totally imaginary. It stems from ignorance, and from having been subjected to propaganda.
Christians have always called Jesus “God.” They never meant that Jesus is the Trinity. Christians have always called Jesus “the Son of God,” by which they never meant that he is the Son of the Trinity. They meant that he is the Son of the Father.
Mary is the mother of God because she is the mother of Jesus, who is God.
To object to calling Mary “Mother of God” on the grounds that she is not the mother of the Trinity is as relevant as objecting: “Mary cannot be the ‘Mother of God’ because fire engines are red.”
And you don't think you haven't been subjected to propaganda from the RCC? Ignorance is not an excuse and being naive is not a virtue. You repeat your nursery rhyme logic as if God can be completely understood and defined and dissected so that you can tell us what God would have us to do with Mary. She is accepted as a member of Christs redeemed. That's honor enough. When you go beyond the bounds of what is revealed in Scripture (Marys sinlessness, Immaculate Conception and eternal virginity), then you trainwreck over the cliff into an abyss of speculation and fantasyland which does not please or honor God.