If you know the jargon, you’ll save your busy server time, and yourself a lot of unnecessarily detailed explaining.
You have four options. Sunny-side up, as you probably know, is an unflipped fried egg, with the yolk still runny, shiny and yellow, and the whites, fully cooked but just barely, still have a sheen to them.
The other three options involved flipped eggs, but what distinguishes them is the degree of doneness.
Over medium: This time, the flipped egg cooks for a minute or two, long enough to partly set the yolk but still leave it a little creamy (yet not thin and runny). If you know what your yolk is like from a 7-minute soft-boiled egg, you know what you get with over-medium.
Over hard: The flipped egg cooks long enough to fully set the yolk, which has the same consistency as a fully hard-boiled egg.