Best omelets I ever ate were prepared on Block Island by a gentleman who had made his living making omelets for 50 years.
He used omelet sized restaurant SS pans, (6 inch), lots of butter, and less egg than most people use. You picked your fillings before the egg was poured. The fillings were kept warm on the back of the stove. Tri folded by flipping both sides with no spatula used at any time. Rolled onto the plate. He used very low heat.
Except for the tri fold flip, That's the way I make them. I use small tin lined copper skillets.
I never did master the tri flip.