How are "Mother of Christ" or "Mother of the second person of the Trinity" not accurate descriptions of who Christ is? When you use "Mother of God" you MUST explain that it does not include the Father and the Spirit. With "Mother of Christ" one MUST explain that Christ is God, etc. What is so special about "Mother of God" as opposed to other descriptions?
As you may know, "Christ" means anointed, or Messiah, not God. Thus, "mother of Christ" doesn't assign Jesus the proper position that He is - God. I do not recall in the Old Testament that makes the connection obvious that the Messiah would be God Himself. Thus, the early Christians would want to make it clear that Jesus was God - and they did so, among other means, by naming Mary the Mother of God.
As to "mother of the Second Person of the Trinity", that is a bit unwieldy, don't you think?
Finally, why on earth are you making such a big deal out of this? Do you enjoy arguing with others?
Regards