Well, Counter Service restaurants aren't "labeled" in any way, but they're easily identifiable in that you enter the establishment, get in a line, order your food from (and pay) a cashier, go to the counter, pick up your order when it's ready, and find your own table. And clean up after yourselves
Table Service restaurants are where you check in at a podium and wait until your table is ready and your name is called. You're escorted to your table and given menus. Somebody takes your order; your food is brought to you; you pay after the meal and - if not on a dining plan - leave a tip.
Eligible snacks ARE indicated by a small purple and white square on the display, I think with the letter 'S' in it.
There are some variations on the above, for example buffets are Table Service; Beaches & Cream is, too, but you might seat yourself - I don't remember.