The origin of Rabbit/Rarebit remains as cloudy as the veil hanging over Snowdon. The "Rarebit" affectation does appear to have been sort of an "elite" adjustment to the slightly insulting original title, a meat substitute for the poor Welsh.
Patrick O'Brian's "Jack Aubrey" provides the fame for a similar dish, "Toasted Cheese", apparently Cheddar melted and browned in a cast iron dish/pan.
The greatest Welsh Rabbit in my memory was my sainted mother's, who mixed some dark beer and a dollop of Worcestershire into the melting cheese, and served it in a flameproof dish atop crisply toasted sourdough slices, the entire dish topped with sliced tomato, then browned under a boiler. Further ascendancy into the pantheon of good eats was achieved by topping the whole thing with a couple of slices of crisp bacon. Mom was a wise housewife, and never purchased "processed" cheese, until the 50s craze for the non-TexMex restaurant version of Chile con Queso, Velveeta heated with a can of Rotel Tomatoes & Green Chiles, arrived. As plebeian as the combination seems (Velveeta??), a dollop on scrambled eggs or an omelet remains hard to beat!