How did a word that meant "food" come to mean "meat, flesh of animals"?

I sometimes work with texts which have an antiquated atmosphere about them, and I occasionally come across the expression to sit at meat in such as

At that time it was degrading to sit at meat with Samaritans.

or

He that serves is greater than he that sits at meat.

I found it explained in dictionaries like the free dictionary or Webster’s. I even checked the etymology of meat as given in the Online Etymology Dictionary:

meat (n.)

Middle English mēte, from Old English mete "food, nourishment,
sustenance" (paired with drink), "item of food; animal food, fodder,"
also "a meal, repast," from Proto-Germanic *mati …

Narrower sense of "flesh of warm-blooded animals killed and used as
food" is attested from c. 1300 (earlier this was flesh-meat, early
12c.).

but I still don’t understand how meat, a word used for food in general, came to mean a particular type of food, flesh of animals used as food.

Edit: I thought it would be interesting to follow the evolution of the use and meaning of this word, if someone had any knowledge or information about this.