I think of a casserole in English as pretty much anything that is cooked in the oven in a single pot combining starch, meat and veg, with or without the lid, cheese optional on top for the "au gratin" (in English, again). In French a casserole is the pot itself, and "au gratin" means made crunchy, not cheesy, on top. You can cook in a casserole (in French) on top of the stove, but a casserole (in English) has to be cooked in the oven, I think, you don't stir it while it's cooking like you would if it were on top of the stove.

Are we making sense yet?