I think it should be about SEO, even though that probably wouldn't matter much in the grand scheme of things. Go for best known name. We don't use William Jefferson Blythe III (or, as you pointed out, Leslie Lynch King, Jr.). Stage name, unless real name was used in a comic. They're characters, not real people. If the best known name includes a middle initial, like most presidents, we should use that, rather than the full second name. In JEB Stuart's case, I vote for J.E.B. Stuart over James Stuart, as he wasn't really known as James or James E.B.
Yeah, the solution's pretty simple for all of the people I mentioned. Nobody ever just says "John Kennedy." It's always "John F. Kennedy" or just "JFK." Same for Robert E. Lee, J.E.B. Stuart, Ulysses S. Grant (who was born Hiram Grant, but changed his name when he was 17,) George H.W. Bush, Harry S. Truman, etc.
UPDATE: Somewhere else, I mentioned that maybe our naming standard for Real People should be "Copy whatever Wikipedia does". Why? Because Wikipedia has already been thru all the arguments about which name to use and has reached a consensus, so why should we re-invent the wheel?
My suggestion has caught on, and has already been used in several Pagename disputes. I don't know if it's official policy yet, but it's getting there....
I'm not saying don't chronicle them, I'm saying put all the chronicles on one page. There are going to be very few real people who actually do enough on (for example) Earth 1 to make a "Real Person (Earth 1)" page really worthwhile. Just bung a continuity header in for each earth and move on.