We have two similar, but different things in our household (American here):

• grilled cheese - slices of cheese (usually cheddar) between two slices of bread; outside is buttered and it’s cooked on a pan so both sides get toasted
• bread w/ cheese (crappy name, I know) - shredded cheese (usually Mexican blend, sometimes mozzarella) on bread w/ mayo to help it stick; either microwave (15 sec or so) or butter the bottom and cook on a pan w/ the lid on (2-3 min?)

Maybe I’ll start calling the second a “cheese toasty” because that’s way better than what I call it.