The third Ecumenical Council, convened at Ephesus in AD 431, dogmatically decreed that Mary is the Theotokos. The Theotokos is a Greek title consisting of two words: Theos meaning "God" and tokos meaning "bearer" or "birther."
The heretic Nestorius taught that Mary was the mother of Christ's human nature, but not the mother of His divine nature. It is true that Christ derived his human nature from Mary alone and did not derive His divine nature from Mary. If Nestorius said only this, he would not have been a heretic.
Instead Nestorius made the claim that Mary was the mother of only His humanity. The problem is that mothers do not give birth to "natures," mothers give birth to "persons." Mary did not birth the human nature of Christ, she birthed the Person of Christ. The Person of Christ is the one and undivided Second Person of the Trinity.
Nestorius' heretical formula essentially creates two Jesus Christ's - the divine Jesus that Mary did not birth and the human Jesus that Mary did birth. But we don't believe in two Christ's. We believe in one Jesus Christ.
The Catholic Pope Celestine and the orthodox bishops rightfully defended the teaching that Mary birthed the whole Person of Christ. Since Christ is a divine person, she birthed God Himself. Hence, she is rightly called Theotokos or "God-bearer."
Now some Protestants that I have met will grant that Mary is the Theotokos or God-bearer but NOT say that she is Mother of God. They claim that "Mother of God" is an incorrect translation of Theotokos.
Are they right? No.
Now the Latin West had translated Mary's title Greek title Theotokos in two ways:
Dei Genitrix and Mater Dei
Dei Genitrix literally means "birther of God." Genitrix, a feminine form of generator, is related to our words genetic and genes. This is the most literal way of translated Theotokos into Latin.
Mater Dei means "mother of God" and it is also very common. It uses a more simple and familiar word - mater or mother. Nobody I know refers to their mother as "my genetrix" or "my genetress" or even "my birther." Mother is more natural. Hence, Mater Dei or Mother of God became the most common Latin title for Mary. It actually predates Theotokos since it is found in the Gospel of Saint Luke. Saint Elizabeth says to the Blessed Virgin:
“And whence is this to me that the mother of my Lord should come to me?” (Luke 1:43, D-R)
The Greek here is Mater Kyriou, meaning "Mother of the Lord." So then, in the tradition of the Church, at the Council of Ephesus and in Scripture we have linguistic and theological justification for calling Mary "Mother of God" and also Dei Genetrix and Theotokos. All three terms generate (pun intended) the same meaning:
Mary birthed one single Divine Person - the Second Person of the Trinity. She didn't birth half of Him, part of Him, or only His nature. She is truly the Mother of God the Son.
Holy Mary, Mother of God, pray for us sinners now and at the hour of our death. Amen.