Elizabeth likely recognized Mary's baby as the promised Messiah. Jews did not believe Messiah would be God - still don't. For this reason, TC's commentary rightly says, "the Bible never ascribes the title "Mother of God" to Mary." It's earliest usage appears to be from Origen in about 250 AD.
By Lord Elizabeth meant Jesus, not the entire Godhead. Jesus is God, but not all of God is Jesus. Consequently the Bible never ascribes the title Mother of God to Mary. She was the mother of Jesus, who was Elizabeths Lord, since He was God. This usage does not necessarily imply that the person using it believed that Jesus was God. Elizabeth apparently meant that Jesus was the Messiah at least. Luke evidently used the term Lord frequently because for Greek readers Christ or Messiah had little meaning. The pagan Gentiles referred to Caesar as Lord Caesar meaning that he was their divine sovereign. Lord had the same connotation for Lukes original readers. Jesus is the divine sovereign for Christians. - TC
Clarification: who’s TC?
Now that's my point exactly. Even if the exact phrase "Mother of God" (or more probably "Theotokos") was not used until 250 AD as far as we know, it's a restating, and a logical corollary, of what was already stated by Elizabeth, and even by the Angel Gabriel.
One can use a phrase that isn't ever used in the Holy Bible. "The Holy Bible", for instance is not found in the Bible. "My personal Savior" is never found in the Bible. "The Blessed Trinity" is never found in the Bible. These are certainly, however, Biblically based truths. Same as "Mother of God."
< The real point here is that one is a mother of a person, not of a nature. Jesus has two natures (Divine and human) but is only one Person --- a Divine Person, who existed from before all ages of ages, co-eternal with the Father and the Holy Spirit. His mother --- which is to say, the one who gave him birth --- not "gave Him existence," but "gave Him birth"--- in that stable in Bethlehem--- was Mary.
Hence she is the Theotokos, the Birth-giver, of Him who existed for all eternity before she did.
As soon as the sound of your greeting reached my ears, the baby in my womb leaped for joy. 45 Blessed is she who has believed that the Lord would fulfill his promises to her!
So she uses Lord to mean Jesus and then uses the same exact word a sentence later to mean something different entirely....God.
Uh-huh.