Well the traditional french way would be that you add spices to eggs before cooking and stuff the omelette right before serving.
I've watched a few youtube videos on the matter and I like how Jacques Pepin explains the process.
Check it here.
Jamie Olivers version is quite similar, he stuff omelette with cheese before he folds it. Check it here.
I've tried all version and I like them all. The main trick about an omelette is that you do not overcook the eggs. Everything else is personal preference I'd say, like whether to chop or no the greens.
Try “massaging” greens w/avocado oil until “silky.” Add to omelette. The oil inhibits production of water from veggies producing a more flavorful, attractive dish.
I use spinach, green onions, red mini bell, fresh ground black pepper and garlic powder.