Previous questions here have addressed the reasons why water or milk is added to scrambled egg mixtures in cooking. However, the two questions I've linked seem to parallel a distinction I've noticed sometimes in recipes or in instructions given by chefs and often repeated on cooking forums: one can add milk or cream to scrambled eggs, while only water is appropriate for omelets, particularly traditional "French" omelet styles.
Some chefs claim that using milk in traditional omelets will make them "tough" (or even "watery"), so they recommend only a small amount of water. (Some traditionalists, of course, also discourage any additions to the eggs before cooking for a French omelet, even leaving any seasonings or herbs until the very end.) On the other hand, I have seen other well-known chefs encourage the addition of milk or even cream to omelets to increase richness.
In my own cooking, I have found that adding any type of liquid can increase the lightness and "fluffiness" of the final omelet, and for that reason it's often worth the extra few seconds it adds in cooking time (which can risk toughness). But I've never noticed a significant difference in toughness when adding milk compared to water, nor have I noticed a significant increase in richness (which I find is often determined more by the amount of butter I put in the pan). In fact, the prohibition against cream seems a bit odd to me, given that many chefs also advocate a rather large amount of butter for omelets which often ends up being stirred into the eggs while in the pan.
So -- my question: is there actually any science behind the claim that one should only add water to a French omelet, since milk (and possibly cream as well) could toughen it? Or is this just another culinary myth? Also, is there any other scientific reason to withhold water or milk from a traditional omelet, other than very slightly increasing the richness of the egg flavor? (Keep in mind that the amount of liquid I'm talking about here is very small, usually no more than a tablespoon for 2-3 eggs.)
(Just to be clear, a French omelet is generally cooked quite fast in a hot pan, often stirred and/or shaken rapidly during cooking to raise the egg temperatures as fast as possible, and then folded and unloaded from the pan while the outside is pale to golden yellow -- never brown -- and left creamy ("underdone") inside. I've occasionally seen chefs advocate a slower approach, but the final product is always the same: barely colored on the outside, creamy to slightly runny on the inside, and the overall goal is to maintain maximum tenderness, often with little or no filling at all. This is opposed to a more "American" or "country" style omelet which can be tougher and perhaps somewhat browned to create a more durable shell used to hold a large amount of heavy fillings.)