Amen, street_lawyer, this is precisely right.
When Catholics say Mary is the MOther of God, we mean exactly what you said, "Mother of the God-Man." We do not mean that Mary is the ultimate "source" or "origin" of the eternal and ever-living God.
This is an except from the Wikipedia, an on-line encyclopedia:
At the Third Ecumenical council, the Council of Ephesus (against the Nestorians), A.D. 431, it was decided that it was entirely appropriate to refer to Mary as the Theotokos, to emphasize that Mary's child, Jesus Christ, was in fact God (Denziger §111a).
That Council clarified that the Church Fathers "did not hesitate to speak of the holy Virgin as the Mother of God" (ibid.), thus affirming what had always been held as true: e.g. St. Ignatius of Antioch, ca. A.D. 110 (Jurgens §42); Alexander of Alexandria, A.D. 328 (Jurgens §680); among other references from similar sources. She is often referred to as "Theotokos" in Eastern Orthodox hymns.
No Christian thinks Mary a "supergoddess" who brings the Trinity into existence. That is not what is meant by the title Mother of God ("Theotokos") It means, precisely, that she is the mother of (woman who gave birth to) a person, Jesus Christ, who is one Being, one Person, God and Man.
Denying mary this title inevitably leads to the heresy of Adoptionism.
Christ gave his life on the cross so that we may have everlasting life.
Romans 10:10 (New International Version)
New International Version (NIV)
10For it is with your heart that you believe and are justified, and it is with your mouth that you confess and are saved.
Romans 10:17 (New International Version)
New International Version (NIV)
17Consequently, faith comes from hearing the message, and the message is heard through the word of Christ.
Ephesians 6:16 (New International Version)
New International Version (NIV)
16In addition to all this, take up the shield of faith, with which you can extinguish all the flaming arrows of the evil one.
2 Timothy 3:15-17 (New International Version)
New International Version (NIV)
15and how from infancy you have known the holy Scriptures, which are able to make you wise for salvation through faith in Christ Jesus. 16All Scripture is God-breathed and is useful for teaching, rebuking, correcting and training in righteousness, 17so that the man of God may be thoroughly equipped for every good work.