You are correct in both cases. The way I explained things is the way these are presented in most general chemistry texts and courses.

When you learn science you also learn that as you progress through courses, terms are further refined. I used to say, by the time I was done with organic, 50% of what I learned in general chemistry wasn't exactly correct. Then you take physical chemistry and find that another 50% of all you knew wasn't exactly correct. The way things are defined in general chemistry are truly general definitions for things, and like you pointed out, there's more too it.