Mary, Mother of God?

Let’s talk logically. I yield to no man in my admiration for the Mother of Christ, but think it is quite a stretch to call her the mother of God. THe mother of a cat is a cat. The mother of a horse is a horse. Are we then to gather that the Mother of God is God? Has Mary ever claimed this title for herself? I have read that, until the 6th century, Mary was called the mother of Christ, then at some council or another, this title was changed. Why?
Not only that, but since we believe in the Trinity, Mary would then be the Mother of the Trinity, as there is only one God. You see where the huge question is here?

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*]Why do we say Mary is the Mother of God?
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