I hope that this is the right place to ask this question. My problem is that I wish to plan a diet for myself but - nearly - every database has different values for the same food. How do I know which one is accurate?
You should generally trust the USDA. They've put a lot of effort into getting everything accurately measured for nutrition labeling, and in the US at least, pretty much everyone gets their nutrition data from them.
That's true in this case: the bodybuilding.com nutrition page says "This database contains data from the U.S. Department of Agriculture, Agricultural Research Service." So there's no good reason they should disagree with the USDA - if they do, I'd assume it's because their data is out of date, or they miscopied it from the USDA database.
That said, the differences are pretty tiny (pretty much insignificant in this case), so assuming they haven't made any big mistakes, you should be fine using either - it's supposed to be the same data. (Personally, I would much prefer the USDA's website or  because they're a lot easier to read than that bodybuilding site.)
Different websites will have different many different factors, including how they calculate their final numbers, such is the problem with any scientific data. For your diet, though, don't worry about it too much. If your diet is so strict that the minor differences in nutrition facts make that much of a difference, then will you have the energy to stick with it in the long term?
Every egg is different, so the chances of the one you eat exactly matching either of those sites is nil. Not to mention that the accuracy of those numbers depends on you measuring ingredients with equal accuracy.
Use the database to learn quick estimates for the foods you commonly eat, so you can make decisions without the computer nearby. If you are out eating, knowing that an egg has about 75 calories and 5 grams of fat, for example, lets you decide how many you want.