The tuna melt is a tuna salad sandwich that has been put through the grilled cheese's discipline, and the heat changes everything about it. Bound tuna salad and melted cheese go on bread that is toasted or griddled in fat, and the defining element is not the tuna but what the heat does to it: the cheese melts down into the salad and binds it, the bound filling firms slightly instead of staying cold and slumping, and the bread crisps against a soft, warm interior. A cold tuna sandwich and a tuna melt share a filling and almost nothing else. The melt is the cooked version, and the cooking is the point.
The craft is in the bind and the griddle behaving together. The tuna salad lives or dies on its ratio of mayonnaise to fish and on enough acid and crunch, celery, onion, a little pickle, to keep it from turning to paste, and that matters more under heat, because warmth flattens flavors that were already dull cold. The cheese is chosen for melt behavior so it flows into the tuna rather than sitting on top of it, fusing the filling into something a slice of toast can carry. The bread is griddled slowly enough that the cheese is fully molten at the moment the exterior reaches gold, the same low-and-patient timing grilled cheese demands: too fast and the bread burns over a cold center. It can run open-faced under a broiler or closed on a flat-top, but either way the structure depends on the cheese setting the loose salad before the bread gives out. A diner can hold tuna salad in the case all day and turn out a hot sandwich in the time it takes to griddle two slices of bread.
The variations stay inside the griddled-melt frame. The open-faced version, finished under a broiler with the cheese blistering on top, is a different texture from the closed griddled build. Swaps of bread, rye or sourdough for white, and additions of tomato under the cheese shift it without changing the logic. The tuna melt belongs to the same family as the patty melt and grilled cheese, all governed by the same heat management, and it shares its cold ancestor with the lunch-counter tuna salad sandwich. Those are their own sandwiches and deserve proper articles of their own rather than being crowded in here.