You can cook an omelet successfully both ways, and other ways too. Julia Child used a fairly high heat technique and cooked an omelet in about a minute making it a practical dinner party affair.
An omelet, like biscuits, appears deceptively simple. They have Very few ingredients, come together quickly to a great reception.
They are foods that I have never seen an adequate description for how to prepare them, even BDL's instructions fail. These foods are all about feel and technique. You have to learn it directly through experience and figure out what works for you and what doesn't.
As an example, I don't like my center to have runny egg as in the classic french fashion. So my preferred omelet technique is non-standard. This means I prefer to flip the omelet in the pan, fill, then roll. I start on medium heat with some stirring toget the egg about half set. I have to leave it undisturbed for a while so it holds together when flipped. Most omelet recipes include fairly vigorous agitation to get most of the egg cooked. Those tend to come apart if you flip them. I'll lower the heat quite a bit while letting it set up. Then I flip, fill and roll, all very quickly. Then cook a bit longer on low to finish cooking the egg and melting cheese if needed.
I cooked four of those this morning for my family and one batch of plain scrambled eggs for my picky eater.