I would say "I will bring a hot dish". That is as opposed to saying that the dish you are bringing will be cold -like a green salad, potato salad, macaroni salad, cole slaw, or platter of cold marinated vegetables. When setting up an event, usually I would just say - "bring an entree" or "bring a side dish" and let them decide if it should be a "hot dish" or cold dish. I don't think I ever asked anyone to bring a "casserole", although I think the terms are interchangeable. I would certainly know what someone meant if they called it a casserole. I have never heard the term "pitch-in" It must be a regional thing.