Eggs – How to add flavor to this amazing looking dish

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.

The sulfur content in the distilled white vinegar used in commercial mayo really emphasizes the egg flavor. If you're using a different vinegar, I think that you should switch to see if it does the trick.

The cooked egg flavor you're looking for comes from pasteurizing the eggs for the mayo. If you've got an immersion circulator, you can do a relatively high-temperature pasteurization for whole eggs in their shell, while not actually cooking them through, and use those for the mayo. There are commercially pasteurized shell eggs available from Davidson's Safest Choice, but they don't taste vastly different from regular eggs— I imagine they pasteurize them at a relatively low temperature for a pretty long time specifically to make them as similar to raw eggs as possible.

Without an immersion circulator, you almost certainly won't have the temperature control you need to pasteurize the eggs without cooking them.

Good luck!