Good question @mfg. I hope I can help a little bit. The basics behind the egg wash are to provide for a couple different thing. These being shine, crispness, and color.
Shine
The shine is primarily provided by the egg yolk. The higher concentration of yolk the more shine.
Crispness
This come from the egg whites. The whites make things a little crispy and sometimes can crackle a little bit (mainly when usually by themselves).
Color
This comes from the fat and protein. Leaving the yoke in along with the shine will add color. You can add water to lighten the color a bit. I find that using the white also dilutes the color a bit but not as much (but the white also makes it crisp). Add cream or milk to get it a little bit darker.
Salt
I often find people adding salt to an egg wash. I does provide some flavor to the crust, but I have noticed that for some reason (when using a whole egg) it does allow it to get a little darker than it would normally.
Sugar
Like salt it adds flavor (sweetness) to the crust. And it will also add some color. Especially if baking at a higher temp and using a raw or brown type sugar.
Alcohol
I have seen the use of alcohol quite a number of times. From the results I have seen it behaves like water. Although not the the same degree. My guess is that it has to do with how fast the alcohol evaporates. As far as flavor, I can't say that it added much at the levels used. I did once definitely get a hint of a bourbon being used in the wash on a pretty plain white loaf. But they did mix in quite a good amount of bourbon to just an egg yolk.
That is all I have really used myself or seen. I am sure that you can use other liquids although I don't know what vinegar would do. Just keep in mind if they are high in fat, protein, or sugar they will add color. Otherwise they will lighten the color.
I have also seen melted butter just used as a wash. It adds a nice color and buttery flavor. I haven't seen it added to egg wash but I would assume you could add to egg yolk to get a shine and probably darker color than cream/milk.
Another important thing to remember is to beat your wash well when using for a bread coating. Unlike if your just using the was was to seal say a ravioli. If your egg was is spotty not one nice fully incorporated mixture your crust can also be spotty.
Best Answer
In my experience you have 3 choices. Lightly spray with water, oil of your liking (olive, canola, grapeseed, or sunflower not vegetable, and lastly, my favorite, brush on lightly sweet unsalted butter. Preheat your over to 350 degrees F (give or take 25 degrees) and do not keep in oven more than 5-7 minutes depending on how round your bread loaf is. Another idea is to slice the bread, spread lightly with unsalted sweet butter and bake for 1-5 minutes in preheated oven 350 degrees F and keep a very close eye on anything you do with all these suggestions because you don't want to end up with croutons. Good Luck and realize I myself would bake my own suggestions starting with less time, let's say 1 minute first, check then add a minute, etc.