I mean more goes into an omelette that just having a fold/flip, but it can still be considered an omelette because the eggs are beaten (otherwise you’re most likely going into scrambled eggs territory), the filling is not mixed in it with the eggs (which would move it more into frittata territory), it’s not relying on dairy (which would make it more like a quiche) and it seems to be thin enough to be folded (if it’s too thick it goes back into frittata or quiche territory). Worst case OP is just making overcooked omelettes.
The only things an omelette purist might bring up is that when the eggs get added you don’t want to cover it up because that lets steam build up which usually leads to a puffy and dry (or drier) omelette. Instead you should slowly stir it to get the omelette to set more evenly. When it starts to set you stop stirring and let it cook a little more so the bottom gets firm (but not like a clear crust) while the top keeps its silky texture. And if you want cheese on the omelette use a cheese grater not slabs of cheese (unless those slabs are very thin) because you’re very likely to overcook the omelette while you wait for the slabs to melt. Of course all of that applies only when you don’t put in too much filling so the omelette stays on the thinner side (easy to fold). If your omelette is already thick due to a lot of filling (hard or impossible to fold) don’t even bother listening to the purists because the purists will tell you to stop butchering an omelette and just make a Frittata. But I say don’t let you dreams be dreams, just do it the way you like it.
