Your beliefs seem to align with what the students are saying and generally with moral realism.

You just said “I think that there are parts of morality that really are culturally relative and subjective, and parts that aren’t.” so you can view some morality as subjective and some as necessarily universal. That is what most people default to and what you seem to saying is wrong with the students. You state they aren’t consistent, but you’re also not consistent. Sometimes subjectivity is right sometimes it’s not. I’m not seeing a distinction, so please elaborate on it if I’m missing it.