I think you are right on with:
So, what does unify the set of stories we're talking about? Seems to me that it's their themes: empiricism, respect for the scientific method (as opposed to the trappings of science, Michael Crichton-style), and a belief in focused introspection. The use of an existing setting serves to play up the contrast between those themes and the assumptions of conventional genre literature, but it's not necessary, and the themes could work just as well without it.
But a thing I think unifies much of it (given that my four closest friends are both rationalists and deveoping authors) is what I like to call "Teapot Smashing":
Teapot Smasing: Activity. Using what you know to get what you want, with expreme prejudice. Comes from the proverb "Any fictional system of physics can be broken like a teapot."
In most of Rationalist fiction, the recurring Ultimate Motivations of the characters is almost exclusively Anti-Deathism and Fun For Everyone. Thus many of the themes involve gaming estabilshed setting rules to enter a feedback loop leading to godhood.
The reason why Fanfic is popular here is that it spares you inventing your own universe to game.
Actually, that might point to a feature that is almost unique to fanfic: protagonists can't deploy the full range of strong munchkin tactics in an original universe without coming off like they've benefited from a deus ex machina, but they can do that in a preestablished universe. Some munchkinry is legal either way, but there are strict limits: you can almost always get away with social, economic, or introspective hacks, for example, because you didn't invent the rules for those. But you can't abuse laws of magic or xenobiology or exotic physics that yo...
This may be old news to some people, especially the Russian speakers, but I didn't see an article about it here.
In 1999, Kirill Yeskov, a Russian paleontologist, wrote The Last Ringbearer, a 270-page take on Lord of the Rings from the point of view of a medic in Mordor's dying armies who is also a "skeptic and a rationalist." In fact, Mordor represents the forces of reason in this retelling of the story. As a Nazgúl (himself a former mathematician) explains, Mordor is "the little oasis of Reason in which your light-minded civilization had so comfortably nestled." Barad-dur is "that amazing city of alchemists and poets, mechanics and astronomers, philosophers and physicians, the heart of the only civilization in Middle-earth to bet on rational knowledge and bravely pitch its barely adolescent technology against ancient magic."
The story has been newly translated and is available in free PDF form -- in English and the original Russian. There's a recent review from Salon as well.