Okay, let me start with terminology. You may think it's incorrect, but it isn't to me. That's why it's "terminology" and not "vocabulary".
Basically, any time you crack an egg into a frying pan, it's a "fried" egg. I don't care what you do with it afterward, I'm calling it fried, even if it manages to become "sunny-side-up", too.
If you crack it (or them) into a bowl and scramble them, then they are "scrambled", regardless of whether or not you keep scrambling them once they are in the pan so that they look like the scrambled eggs at the breakfast buffets (but, hopefully, yours taste better). So, yes, much of what I call "scrambled eggs" are "omelettes" to others, whether or not they have something in them.
Also, when I put something into an omelet, I tend to start cooking that in the pan first. When everything is ready, I pour the eggs in last (unless I'm adding cheese, because that's almost always last). This is in contrast to buffet that have cooks that make a blank egg and then just throw everything in the middle and fold it up. I hate that, but what are you going to do.
Well, I did that this morning.
Rather than make a breakfast burrito, I made eggs in the shape of a tortilla, and in another pan cooked up half a hamburger patty (crumbled) with a little salsa. I was careful not to get too much juice. Then I peeled some Monterey Jack cheese. Yeah, I used a potato peeler because the grater breaks up the block. I guess because it's too soft.
After flipping the egg, which was easier than I expected in the bigger pan, I flipped it onto a plate, added the contents, topped with a little more salsa and cheese, and then rolled the egg almost (but not quite) like a burrito.
Lessons Learned: First, sometimes you just have to dirty (and wash) a second pan, as much as I'm loathe to do that.
Second, next time try a bigger pan. The omelette was too thick for use as a tortilla.
Third, this was a really silly idea because the egg was too frickin' hot to roll and wrap!