If Jesus dies Jesus is the second person of the trinity. Jesus dies on the cross in his humanity. It is also because He is one person, that God is said to have died on the Cross for sins.
This is because within the actions of one person the realities and values are communicable. The value of the action of Jesus as God-Man to suffer and die for sins is why you can say in both ways that are true. A Man died on the cross and God died on the Cross.
If there is no communication of values in the one person Jesus, there is no value in the sacrifice. He was always Fully God and Fully Man.
The value of Motherhood of Mary as Mother of God does not require her to be some kind of author of God. She only has to be the Mother of Jesus. But the Title Mother of God inherently teaches that God was Her Son, and that Jesus Is Was and Always will be the incarnate God.
Does that answer it for you?
No, it doesn't because the word *God* means more than and is used for more than, only Jesus.