This is what I'm thinking:
pm. "cannot but, have to" this seems colloquial, and the fact that dictionaries think of this as a set phrase/fixed expression makes me think so. Also feels like the "can't help but" of 忍不住.
adv. "must, have to" this feels to me like it might be formal, is that right?
adv. "have to, have no choice but, be forced to" this sounds begrudging, like you have to or must, but don't really want to.
aux. "should, ought to, must" I think I usually hear this more often to mean "should, ought to" and not really as must, but I'm not that sure.
Can anybody clarify where I'm right and where I'm wrong in my thinking?