The tuna melt is not a cold tuna sandwich with cheese on it; it is a toastie built over a tuna filling, and the seal is what defines it. Bound tuna mayonnaise goes between two slices of bread, cheese over the tuna, and the whole thing is pressed in a hot iron or run under a grill until the cheese melts into the fish, the outside crisps, and the edges weld shut. The heat is the entire transformation. Cold tuna and cold cheese is one sandwich; the same components fused by a grill, the cheese gone molten and bonded to the warmed fish inside a crisp shell, is a different one with its own name and its own behaviour, which is why it sits with the toastie family rather than the tuna shelf.
The craft is heat management against a wet filling. Tuna mayonnaise carries moisture the way a plain cheese toastie does not, so the bind has to be kept tight, drier than a cold tuna sandwich would take, or the steam it releases under the grill softens the bread before the outside has crisped and the seal never forms. The cheese is chosen to flow rather than split, usually a strong Cheddar, and laid over the tuna rather than under it so it melts down into and seals the filling instead of sliding off the outside. The heat is moderate and patient: too fierce and the bread scorches over a filling that is still cold in the centre; too slow and the bread dries out before the cheese has gone molten and the tuna has warmed through. Buttering the outside of the bread is what gives the crisp, lacquered shell that holds the hot, soft centre together when the sandwich is cut.
The variations stay inside the pressed-and-melted frame rather than wandering back to the cold shelf. Sliced tomato laid under the cheese is the most common addition, bringing acid and its own steam the seal has to contain; sweetcorn worked through the tuna survives the heat and adds pop; a slice of pickle or a stripe of hot sauce cuts the richness of fish and melted cheese together; an open-face version finishes under the grill rather than sealed in an iron. Each deserves its own article rather than being crowded in here.