The cheese toastie is the reference toastie, the one every other sealed-and-pressed sandwich is a variation of, and it is defined by the seal. Two slices of bread are buttered on the outside, filled with cheese and nothing else, and pressed in a hot iron or under a grill until the outside crisps hard and the inside goes molten and the edges weld shut. That weld is the whole sandwich. It traps a soft, hot, slightly liquid centre inside a crisp shell and stops the cheese escaping when the toastie is cut. A toastie that has not sealed is just cheese on toast folded over, which is a different and lesser thing.
The craft is heat management against the clock, and it is unforgiving because there is nothing else in the build to hide behind. Too hot and the bread scorches dark over cheese that is still cold and rubbery; too slow and the bread dries out and stiffens before the centre ever flows. The butter on the outside is structural, not seasoning: it is what conducts the heat into the crust and what browns it evenly, and a dry outer face will toughen rather than crisp. The cheese is chosen to melt rather than split, typically a strong Cheddar whose sharpness survives the heat, and the load is kept light because an overfilled toastie blows its seal and bleeds. The press matters as much as the heat, since the pressure is what fuses the two faces into one closed object rather than a hot sandwich that falls open.
The variations all stay inside the pressed-and-melted frame and change one thing each. The jaffle is the same idea made in a stovetop iron with crimped, fully sealed edges; the ham or onion toastie adds a filling that is also water the seal has to contain; the tuna melt runs a bound tuna under the grilled cheese; Welsh rarebit is the open relative, a thick cheese-and-ale sauce grilled until it blisters, closer to a sauce than a slice. Each deserves its own article rather than being crowded in here.