Good questions, to which the answers are complex and much more in-depth than I would like to go in this thread.
For now, let's just discuss the specific case from the OP of her daughter resisting correction with a threat: "If you say that, I'll say 'I won't like the piano anymore'"
In this situation, the mom starts off playing the role of TEACHER. The daughter doesn't like the corrections and doesn't want to do the work, so she responds as a DAUGHTER with the threat to quit piano. So, now the mom has to decide how to address the manipulation: whether to (1) use reverse phsycology, (2) say 'OK, fine, just keep going then', (3) get tougher and insist on obedience, or (4) stop the lesson altogether and have a serious discussion about piano lessons.
As both a teacher and a parent, I would imagine having two completely different reactions to such a statement depending on who was saying it to me (either a student or my own child) and my strategy in dealing with it would be completely different in either situation. So, what if I'm both? What do i do? I'm not saying it's impossible to teach your own kids. Like you said, you are always teaching your kids all the time anyway. I'm just saying that it is more complicated than it is with any other student.