Thanks for the input. I agree that your teacher introduced that concept poorly, from the description you've given. My advice applies only when the name being introduced is strongly tied to specific behavior.

It sounds like your trouble was that the name "normal" did not come attached to specific behavior, so the appeal to intuition at the end failed. I imagine that if the teacher introduced "cyclic" groups using this pattern, that may have been less opaque.

Even then, there's a difference between "thingies of this type are the same as thingies of that type, so we can call them by the same name" and "once we've shown this property we're allowed to use this name, which pumps your intuition".

In fact, this suggests a fun way to introduce cyclic groups -- mention that you want to define a "cyclic" group, talk about the properties you're going to need to justify the loaded name, guide discussion towards generator objects, formalize the idea, and then show how generator objects work with infinite groups. Might be a little less unnerving than learning that Z is cyclic right off the bat.

Even with these caveats, I readily acknowledge that this method of teaching won't work for everyone.