RAREBIT, n. A Welsh rabbit, in the speech of the
humorless, who point out that it is not a rabbit. To whom it may be solemnly
explained that the comestible known as toad-in-a-hole is really not a toad,
and that riz-de-veau à la financière is not the smile of a
calf prepared after the recipe of a she banker.   -
"Welsh Rabbit" predates the "Welsh Rarebit" in the literature. The word rarebit appears in no other context, thus was certainly made up by some clueless scribbler.
See Welsh Rabbit.