In my language, which is very distant from the english/frank dicotomy, we never had that distinction between peasant and nobility food and still we get a lot of words to distinguish between meats.

Mutton is either carneiro/ovelha (ram/sheep) or just chanfana (this word is used to denote the meat comes from a fully matured animal, over a minimum of two years old, usually around four or five).

Lamb can either be borrego (most commonly used word) or anho (a less used word, alledgely tied to the time we were under moorish occupation).

The words are imperative and not there to make things more palatable; these can't be thrown around to designate the meat solely, as the meat designates the animal and vice versa.