My understanding is that the difference in terms goes back to the Norman invasion, which is when a ton of French-based terms for things were carried over.
The peasants referred to everything as the name of the animal but the French nobles referred to it as porc, boeuf, etc. This is also where we got the words for venison, mutton, veal, poultry, and also apparently pheasant
To add to this, the rich (i.e., French-speaking) consumed the most butchered meat, by far. So, it came to be that butchered meat for sale would be labeled in French, while the live animals, which were tended by (English-speaking) peasents retained their English names.
Buffalo in english is a weird word, because it's an animal, a city and an action, which is why the phrase "Buffalo buffalo buffalo buffalo buffalo" is one of the weirdest things I've ever heard, but completely correct english.
Piggybacking off of this, "venison" comes from a Latin word meaning "to hunt" and was originally used as more of a catch-all term for game meats. You might have deer venison, boar venison, rabbit venison, etc. Over time it came to mostly be used to refer to deer
From what I can work out, yep it seems that way. Pork and beef were too expensive for the peasants so they just referred to them as the animals they were raising, but chickens were actually on their menu so we ended up keeping the animal words for it. We still got the word for pullet (young hen) though.
I just read a theory that poisson, french for fish, didn't come over because it sounded too much like poison, but who knows if that's true lol.
If you don't mind my asking, which language is yours?
It's an interesting question to ponder which different languages ended up with distinction words for the meat vs the living animal, and maybe what that says about the culture.
The distinction is not a feature of French, from what I understand, and English ending up with this distinction seems to have been entirely accidental.