Omelette cooked in oven or pan?
Just because the recipe calls it an "omelette" doesn't make it one. An omelette starts and ends its cooking in a pan. If you finish it in the oven, it becomes a frittata. (There are a couple of historic differences, but let's just stick with that.) If the egg mixture both starts and finishes in the oven it would probably be a crust-less quiche.

The benefits of cooking in the oven is the ability to get more even cooking of the eggs to get a more consistent texture throughout.