I’d rather prefer a systematic approach where counter is the feature and an additional tag says what it sells. I’d maybe also distinguish cold cuts from deli, but less sure on this.

As a native English speaker I have never heard the term cold cuts, I assume it is an American term for the deli-counter?

A deli-counter is where loose cold cooked meat, cheese and other cooked items such as sausage rolls and pork pies are sold.