In my experience if they're semicircular it's because they're folded over on themselves. Not sure what that has to do with cooking (or not) both sides.

Your ingredient mixing distinction doesn't reflect what I've encountered. That seems to have more to do with the nature of a given ingredient or alternatively with presentation or other concerns specific to a given recipe.

Yeah if you wanted "cooked separately and also circular" you'd need to make a two omelette sandwich. I've yet to encounter that at a restaurant.