Water vs. milk/cream (or nothing) in traditional (French) omelets
Americas test kitchen has tested this in a prior issue.  I cannot put my hands on the issue right now which explained their process and the results.  It also contained a good deal of explanation about the science of the process. They tried all sorts of liquids in their omelette and scrambled eggs recipe. They found that water did make the eggs slightly fluffy but did not slow the coagulation process. This led to a tougher scramble than milk. They wanted the same result for an omelette with a denser texture.  The end result being that butter was better for an omelet.

However, some of the result was later reported in a Smithsonian Magazine Article.  It is a much more compact summary but fairly on the nose for your question. 

Their answer was as follows:


  Add milk to scrambled eggs, frozen butter to omelets: If you want
  scrambled eggs, most of us know to throw in a bit of milk or butter
  while scrambling. That’s because the lipids in the dairy coat the
  proteins in the egg (11 percent in the whites and 16 percent in the
  yolks) and slow down the process of coagulation, a.k.a. when the
  proteins are denatured and unfurl, releasing much of the water in the
  mixture. Adding fat helps keep some moisture in and fluff up the final
  product. But the same does not go for omelets. “While scrambled eggs
  should be fluffy, an omelet is more compact,” the authors write. 
  While milk works for scrambled eggs, it can add to much moisture to an
  omelet. The chefs recommend frozen bits of butter instead, which melt
  more slowly and disperse more evenly. And it turns out you can go
  ahead and salt the eggs before you even cook them up. Because salt
  affects the electrical charge on the proteins, it weakens the bonds
  between them, preventing overcoagulation. Bring that up at your next
  brunch.


The Science of Good Cooking: Tips From America’s Test Kitchen