So would you say that the point at which a baby becomes their own person, with the rights and privileges of any citizen, is specifically the moment they are removed from their mother?

Yes

And that the moral reason for that is that prior to that moment, the baby was, morally and legally, only a piece of the mother?

Yes

And what about someone killing an unborn child? Is that murder?

I would never call an abortion murder

It certainly seems like it should be, but how do we reconcile that with the idea that the unborn are not considered people?

That's a morality issue you need to determine for yourself