I would call it a custard, I think, if not only for want of a better word. A custard consists of milk, egg and a thickening agent i.e. flour, so it fits the description well enough.
It might sound a bit odd to English-speaking ears though. I think 'savoury custard' is probably more appropriate. I must admit I always thought moussaka et al was topped with a white/bechamel sauce; I'm pretty sure that's the more common topping in 'the West'.
As for the dishes themselves, I'm not sure if there is a catch-all English term for them - 'pasta bake' is an English version of the Italian pasta al forno, but that obviously only applies to dishes that use pasta.
I have a theory about your omelette (and sadly, no means to test it). But I think it is plausible, please feel free to point out logical errors.
I think that the unappetizing coloring is due to the creation of sulfur salts in your pan. Egg whites are rich in sulfur, as are all alliums (including chives). If you heat them enough (you don't mention whether you overcooked the omelette, but it happens often enough), the heat changes the molecules in which the sulfur is bound, and the sulfur atoms (or sulfur containing ions) are free to react with whatever they find around. If they happen to find iron or copper, they form sulfides and sulfates with them. These sulfides and sulfates have colors ranging from blue to green, and pure iron sulfide is black. A mixture of them with the yellow yolk can easily look gray, especially when the color is desaturated through the addition of white cream cheese.
You say you used a coated pan, but there are still possible sources for the iron and copper. First, nuts contain trace elements, including iron and copper. Second, yolks too contain iron (in fact, the bluish coloring on the surface of a hard boiled yolk is an iron sulfide created when the proteins in the egg white denaturated from the heat). Third, you can't exclude contamination of the ingredients (plants sometimes store metal ions from the environment).
I have no way to prove any of the above, but at least it sounds like a good working theory. But I will understand if you aren't eager to reproduce in order to investigate the effect closer.
Best Answer
I have heard something similar to what you're describing called a "scramble". It's usually what I wind up with when I'm out of practice with making omelettes.