If He is one person, then saying that Mary is mother of GOD says that Mary if the mother of deity.
This is where the misapprehension lies. There is nothing in "Mother of God" that's opposed to "Mother of Jesus."
Like good judges, we should take care to look into the legislative history. You with me on this?
The term "Mother of God" came out of the Council Ephesus (431 AD)specifically to uphold he Biblical rtuth about Jesus and refute the Nestorian heresy. Jesus possesses fully the nature of God and the nature of man: we can describe Him as true God and true man.
He has a human body and a human soul. With His brain He thought just as other humans think. With his will and His emotions he desired, and chose, and loved and felt just as other humans do. Scripture says that He is human, like us in all things, except sin.
That is His human nature. Because He is a Divine Person,the Eternally existing Word and Son of the Father, equal to the Father and the Spirit in all things, He also has a divine nature. He is One Divine Person with two natures.
If Jesus was two different persons, then the Incarnation didn't happen. OK. No controversy so far, I hope.
Nestorius did not get it. He believed that Jesus Christ was "part" man. He believed that that Mary gave birth to Jesus, a being who was not-quite-human, separate from the divine Logos (a different person) but sort of possessed by Him. Nestorius thought Jesus did not have a human soul, so that in a sense he was a puppet. He was only wearing a "human costume," He was pretending to be human.
Nestorius didn't seem to grasp that in the absence of a human soul, Jesus would be a simulacrum, a zombie, in Hebrew terms a golem, less than human. This denies the reality of the Incarnation.
It was Nestorius who proposed that the long-used title for Mary, Theotokos ("God-bearer") --- derived from Elizabeth's greeting ("Who am I that the mother of my LORD should come to me?") --- was wrong, because the "man" Mary bore was neither really Man nor really God. It was Nestorius who proposed to substitute an innovation, a term he had coined, Christotokos, Christ-bearer.
The Council found him wrong because they saw that this was a rejection of the Incarnation, the fact that there is one Divine Person who possesses two complete natures, and that Jesus does indeed have a human soul (with human intellect and will) as well as the Divine attributes of infinite intellect and will.
Since any mother gives birth to a person (and not just a "nature,") Mary is the mother of the Person Jesus Christ, God and man.
It means she carried Him, Jesus Christ our God, in her womb and gave birth to Him. It does NOT mean she is "older than God" or some sort of Mothergod or goddess, or the "Source of the godhead" or "Mrs. Trinity" or that she somehow parthenogenically generated divine qualities or any such nonsense.
Just Google "Council of Ephesus," and you will see the legislative history. The whole intent of the title "Theotokos" is to safeguard the Biblical fact that Jesus is one Person with two complete natures.