Honestly, the argument over calling Mary the Mother of God is just stupid. She was the mother of Jesus who was God hence Mother of God. Really, this stuff isn’t too difficult. It’s really just the Transitive Property of Religion: If A=B and B=C, then A=C. Unless of course, we all disagree that Jesus was God.
You have not completed the final inference that your line of thinking is meant to imply. Here is what you are trying to say:
"Mary bore Jesus; Jesus is/was God; therefore Mary, the Mother of God, is also Divinity."
That, of course, is not true, and the Islamists know so, but they want to perpetrate the false idea that their Allah is the God of the Bible in the credulous minds of the ignorant, who have made Mary out to be a Co-Redemptrix with Jesus, and therefore equal to, and to be worshipped as, The God.
I reject your linguistic twisting of Elizabeth's address to Mary, as does Albert Barnes, whom I quoted.