“Yet Luke did not write mother of Christ which would be mother of Messiah but mother of my Lord.”
If you wish to understand the text, you must look at the context and understand that “Lord” was used to refer to Messiah. This is Who Mary was informed she would carry and why she was blessed.
Clearly, Mary was not Mother of Yahweh - the other option for “Lord.”
Best
I can only repeat the previous explanation: When Elizabeth was speaking and when Luke recorded it, they chose the word that is a direct reference to God according to the language in use. No different than when I call Mary Mother of God today. If you can infer that God in Elizabeth’s speech is the Christ, you should as easily infer that God in my speech is the Christ.