Moderate heat, eggs at room temperature, non-stick egg pan (8" is good, with gently sloping sides) with a tight lid. Melt butter in the egg pan until it stops foaming. Crack your eggs into a bowl so you've got more control when you add them to the egg pan. Cook uncovered until just the bottoms of the white are set, the tops of the whites should still be transparent. Add 1Tbs water for two eggs, cover, check after 45 seconds and give the eggs a jiggle to be sure they will slide neatly out of the pan. They may be ready at this point, they may need another 30 seconds to a minute.
EDIT: I'm hungry anyway:
My pan isn't ideal, I'd rather have 8", this one is 10" but it has a lid. My eggs are at room temperature, I have butter ready and a tablespoon of water. A heat-proof rubber spatula is nice to have, but if everything goes right you don't even need it. I turn the heat to 1 click out of 10 below medium (YMMV [Your Mileage May Vary]). After giving the burner and pan a couple of minutes to heat up it should take room temperature butter about 10 seconds to fully melt and another 5 seconds to stop sizzling.
One thing that I like about cracking eggs into a bowl first is that it allows me to better center the yolks.
The whites should start to become opaque almost immediately, but they shouldn't take on any brown color or be sizzling hard, just gently. Once the bottoms of the eggs are completely opaque (this should take less than 30 seconds), add the water and cover immediately. Now the eggs are steaming.
Fortyfive seconds and a jiggle later they're just about there! (This is when I'll add salt and pepper, I'll wait until they're plated for the hot sauce I can't live without.)
10 more seconds under the lid, slide onto the plate. Voilà, completely set whites, completely runny yolks. OOPS, I forgot the tiny bit of white on top. That's easy, using a spoon, give the eggs a quick baste of the hot butter before adding water and placing the lid.
You can also just steam the eggs until the yolks have a bit of opaque white on top with or without basting them. With practice you can control just how done you get your yolks on a scale that runs between totally runny, to creamy, to still translucent looking but nearly solid, to dry and crumbly (after piercing the yolk, usually).
Flipping is a different technique to get a very similar result.
One more quick note: You may notice that I used quite a bit of butter. That's just personal preference when my jeans are fitting just fine. You may use much less butter if you prefer, another type of fat, or even just a spray of non-stick spray (Pam). If you use a minimum amount of fat, or even none, the "check and jiggle" step becomes even more important. Make sure you can slide the eggs around in the pan. If they're sticking at all, this is when the heat-proof rubber (silicone) spatula can be very helpful.
Best Answer
The quote in your question is clearly aimed at eggs that are mixed (ie omelettes, scrambled eggs etc) before cooking, not for whole eggs sunny side up. The benefits of adding more water is to slow down the cooking of those mixed eggs.
If you add a liquid, water or milk, you'll be poaching the eggs rather than frying them in fat or oil as most sunny side up eggs are cooked.
If poaching is what you want to do, there's plenty of instructions on how to do this out on the web. Poached eggs are equally as good if not nicer than fried 'sunny side up' ones in my opinion.