A toastie is a category before it is a recipe. The word names a method, not a filling: any sandwich that has been buttered on the outside, filled, and pressed hot in an iron or under a grill until the bread crisps and the edges close. What makes something a toastie is not what is inside it but that it is sealed and pressed and served hot, which is why the word stretches across cheese, ham, beans, tuna, and a hundred other fillings without ever needing a new term. The toastie is the British shorthand for the whole sealed-and-pressed family, and the family is defined by the closure rather than the contents.
The craft that the word implies is constant whatever the filling. The bread is buttered on its outer faces because that fat is what conducts heat into the crust and browns it; an unbuttered toastie toughens instead of crisping. The filling is kept modest because every wet addition is moisture the seal has to contain, and an overloaded toastie blows open along its edge and bleeds onto the iron. Heat is managed against time: hot enough to crisp the outside, slow enough that the centre is properly hot before the bread scorches. The press is what fuses the two faces into one closed object, and a toastie that has not been pressed and sealed is just a hot sandwich that falls open when cut. Those rules hold for the cheese build and equally for every other thing the word covers.
The variations are the fillings the format absorbs, and the format absorbs almost anything. The cheese toastie is the reference case and has its own account. Beyond it sit the ham and cheese, the cheese and beans, the cheese and onion, the tuna melt run under the grilled cheese, and the jaffle, which is the same idea made in a stovetop iron with fully crimped edges. The shop versions and the toasted sandwich maker that put the method in every kitchen are part of the same story. Each deserves its own article rather than being crowded in here.