"My bet is on Archangel Gabriel."
You are right that the term Theotokos (which again, is specifically *not* the same term as "Mother of God" in Greek -- it is the Birthgiver of God, or Bringer-forth of God -- hence we Orthodox in the English speaking world leave the word untranslated) came long before the council in Ephesus.
Nestorius's avoidance of the term was odd enough within the Church that it drew attention to him, even though he had mostly kept his heretical beliefs to himself.
And while the Archangel may have used the term, we do know that something very close was used by St. Elizabeth, who called her "the mother of my Lord."
I’m curious as to where precisely the term "mother of God" comes from. You’re absolutely correct that the Church technical meaning of Theotokos is "the birthgiver of God". This is a far different meaning than what is being applied here as the "mother of God" which implies far more grandeur. This is a subtle but important difference.
Nestorius clearly had a wrong perception of the essences of Christ, believing Christ was two seperate entities, human and divine. He objected to the term "birthgiver of God" (Theotokos) instead favoring the term "birthgiver of Christ" (Christotokos). But his was not an isolated view as you would seem to dismiss. His view was enough to cause the Chalcedonian schism and the separation of the Assyrian Church of the East from the Byzantine Church. According to my readings, today it is believed that Nestorius and the Church of the East's views were exaggerated.
However, Nestorius isn’t the point. The point is whether Theotokos can be accurately translated into “mother of God” which you have rightfully stated is an inaccurate interpretation. I have not yet been able to find when the actual term "mother of God" first appeared as the writings seem to incorrectly use these interchangeably; but it is an imprecise and inaccurate definition of the meaning of Theotokos. With all due respects, the Church seems to fosters this misconception by claiming the Theotokos and then substituting the different term "mother of God".
I wouldn't bet on the Archangel Gabriel using such a term because it is not recorded. As you, yourself pointed out, Elizabeth did not even call Mary, "mother of God" but properly stated, "...mother of my Lord". This is consistent with the precise definition of Theotokos. Mary was the mother of MY Lord. Mary was the birthgiver. Nothing more. A view that was correctly held by the early Church but that has now become distorted.
I've noticed there was no response on my question to whether Mary was sinless in direct contradiction to Rom 3:23.