And, rarebit (while your derivation is right!) has nothing to do with real rabbits. It's a thick cheese sauce on toast. Why Welsh rabbit? Don't know but probably it was rude.
I looked up a recipe since haven't had this in forever and forgot what's in it - cheddar in a cream sauce with a little hot sauce, mustard, etc. - made me hungry. One of the best comfort foods ever.
It's the same with Bombay duck, has nothing to do with a duck, it's dried fish (I know cos it's one of the national dishes of where my family is from)
From what I understand, rarebit is somewhat of an ironic term. Something to do with the welsh being really poor not being able to afford butchers meat, and in England, rabbit was considered poor mans meat and so in Wales, cheese was considered poor mans meat.
It's kind of a stab at the English to say "haha we don't need rabbit meat to enjoy a good meal. Our cheese is our rabbit"
In the welsh language, rarebit means baked cheese.