The article you linked claims "any type of omelette", but the vast majority of omelettes[*] are semicircles, not circular, right? You'd have to cook the top and bottom separately or mostly separately to get a circular one. Hm.
[*] of course, here I mean proper omelettes, which are an egg shell around ingredients, not scrambled eggs with ingredients mixed in.
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.