Would it be correct to say you are differentiating between giving birth to Christ and giving birth to True God, God Incarnate - and that Mary is more aptly the "Mother of Christ"? No. She is not the "Mother of Christ" bearing the "Messiah". Scripture states that she was the mother of our "Lord". Christ also stated, "Who is my mother...these are my mother...who do the will of God." illustrating the same level of notoriety with Mary as with any woman.
Please note the following:
Luk 7:28 For I say to you, Among those who are born of woman there is not a greater prophet than John the Baptist. But he who is least in the kingdom of God is greater than he.
No one born up until that time was greater than John the Baptist. According to our Lord, the John the Baptist was greater than Abraham, Moses, David, etc. Presumably he was also greater than Mary having been born after Mary.
"she was the mother of our "Lord"…Jesus Christ, Son of God, the Word become flesh...
If you mean to say by "Lord" the same as God Incarnate; God, The Son, then I don't see the difference here. If not, aren't you introducing a fourth person?
Do you see my point?