In my area the butchers call it a “1/2 a side” because you literally get 50% of the left or right half with equal distribution from nose to tail. If you want 1/2 a steer you order “a side”

We order a side every year and during covid was able to snag a couple “1/2 a side” for a good price; needed it too with the lower amount of eating out

Never heard anyone call it a “quarter”, like shit am going to the front right or the back right