The Welsh rarebit sandwich is defined by its sauce, not its cheese. A plain toastie melts a slice of cheese until it runs; rarebit cooks the cheese first into a thick, savoury paste with ale, mustard, and Worcestershire beaten through it, so what goes on the bread is already a seasoned, emulsified spread rather than a slab waiting to soften. That is the whole distinction. The sauce carries a bitterness from the ale and a sharp back-note from the mustard and Worcestershire that no sliced Cheddar between bread will ever reach on its own, and grilling it until it blisters concentrates that flavour rather than just loosening it.
The craft is in the sauce and the grill. The cheese is melted gently with the ale and seasonings so it thickens to a paste that holds its shape on a knife rather than splitting into oil, because a broken rarebit slides off the bread and burns instead of blistering. The toast underneath is taken firm and dry first so it can carry a wet, heavy topping without going soft, then the paste is spread to the edges and put under a hot grill until the surface bubbles and browns in patches. As a closed sandwich the same paste is laid between slices and the whole thing is grilled or pressed, but the sauce still has to be cooked thick beforehand, because raw cheese and ale between bread is a different and lesser thing.
The variations are codified rather than loose. Crown a finished rarebit with a poached egg and it becomes a buck rabbit; some cooks build it on a base of ham or with a slice of tomato under the sauce; the plain cheese toastie is the stripped relative that skips the cooked sauce entirely and relies on a slice melting in place. Each deserves its own article rather than being crowded in here.