omelette (also omelet) a dish of beaten eggs cooked in a frying-pan and served plain or with a savoury or sweet filling [French omelette, obsolete amellete by metathesis from alumette variant of alumelle from lemele knife-blade from Latin lamella]. Concise Oxford Dictionary.
The key thing is the beating. After that, how that egg mixture is cooked can be very different...
In Thailand, an omelette is ไข่เจียว (kai jiaw), where เจียว is a variant of 'to fry' in that a stirring motion is used (but is not scrambled), rather than ไข่ดาว (literally, egg-star) which is a fried egg (to be placed on top of some fried rice). A Thai omelette has crispy, bubbly, golden edges where the egg mixture has been cooked in quite a lot of very hot oil in a very hot wok so it bubbles and spits on entry. (Or, it can also be cooked and folded in a large, soft pillow.) The technique, ingredients and quantity of oil to perfect this, is more difficult than it sounds and needs (I imagine) several years of practice to get it golden brown (not burnt), bubbly (not dense), crispy (not soft), and without too much oil clinging to the finished omelette. But when done perfectly, using plump crab meat for example, served with soft and steaming white rice with a generous splash of prik nahm pla (พริกน้ำปลา), it is a dish of simple complexity.