I have a specific memory of being informed it was a misconception of non-Catholics that the "immaculate conception" was Jesus, not Mary. Her conception was immaculate because she had to be pure to be blessed with carrying the baby Jesus later in her life. I can't recall if this was in "New Testament" or "Catholicism" class. It wasn't yesterday to be sure.
i think you are right that the immaculate conception (ie, without sin) refers to Mary's birth, while the virgin birth is Jesus's birth. i think that these are 2 distinct events. (however, i am reporting from memory of a conversation i had about twenty years ago.)
Once again with feeling...The Immaculate Conception is a dogma which states that Mary was conceived without sin on her soul. This is not related to the physical part of her conception through the cooperation of her parents, the holy couple Anna and Joachim.
For some reason, people keep making the mistake in thinking the I.C. means something about having a baby without sex, hence they get it confused with the birth of Jesus. Jesus is the Incarnation, that is, God sending his son to take on a human nature. This happened through the power of the Holy Spirit, not anything to do with intercourse. It's a mystical thing. That is why Christians' belief in Jesus as the Son of God requires faith!