[Is it MORE specific to say that Mary gave birth to Jesus Christ,the incarnate 2d Person of the Godhead, OR is it more specific to say that Mary gave birth to God?]
Mary was the virgin God used to conceive by the Holy Spirt the incarnate God in the flesh, Jesus of Nazareth who alone was crucified for our sins,died and rose again the third day so that those who believe and confess Jesus as the Christ of God may be saved by grace throught faith without works according to God's Word, the Holy Scriptures.
Catholism abandoned the Christ only salvation gospel some centuries ago and is now reprobate concerning the Christian faith and has overun by mysticism and heresies, as the Holy Bible warned against.
In the book of Galations chapter one, God warns Christians concerning the Word of God and His Christ;
1 Paul, an apostle, (not of men, neither by man, but by Jesus Christ, and God the Father, who raised him from the dead;)
2 And all the brethren which are with me, unto the churches of Galatia:
3 Grace be to you and peace from God the Father, and from our Lord Jesus Christ,
4 Who gave himself for our sins, that he might deliver us from this present evil world, according to the will of God and our Father:
5 To whom be glory for ever and ever. Amen.
6 I marvel that ye are so soon removed from him that called you into the grace of Christ unto another gospel:
7 Which is not another; but there be some that trouble you, and would pervert the gospel of Christ.
8 But though we, or an angel from heaven, preach any other gospel unto you than that which we have preached unto you, let him be accursed.
9 As we said before, so say I now again, If any man preach any other gospel unto you than that ye have received, let him be accursed.
10 For do I now persuade men, or God? or do I seek to please men? for if I yet pleased men, I should not be the servant of Christ.
11 But I certify you, brethren, that the gospel which was preached of me is not after man.
12 For I neither received it of man, neither was I taught it, but by the revelation of Jesus Christ.
You shouldn't bear false witness against Catholics in that manner.