I agree with your first statement, but I'd point it the other way around. I often call big supermarkets grocery stores even when I shouldn't. And I never use "superstore" or "box store".

I'm fine calling walmart a supermarket.

Language is clearly missing something if we don’t have a different word for Walmart, Target, etc. versus Albertsons, Fry’s, etc. I think if the latter set are grocery stores, then the former are clearly something else.