Good question. I've been thinking of the social arm of the rationality scene for a while as a revival or modernization of the concept of a fraternal society, of which the Masons are probably the most famous example, or of the social clubs with which they were often associated; but I haven't looked much into best practices for that sort of organization. In retrospect it seems like a stupidly obvious thing to be asking. Here's what I can think of off the top of my head.

Fundamentally I think societies like the Masons get most of their value not from any specific goals or practices, but simply from being a nucleus for cooperation and social contact among sub-Dunbar groups of intelligent people. As long as you only care about intellectual interaction, it's probably easier to find such a nucleus now than it was in the decades when these groups were most prominent -- the whole point of social media is to provide a platform where groups of people can agglutinate -- but if you're looking for physical, in-person interaction it's harder than ever. Any further development of this aspect of the scene should keep that in mind.

Having somewhat nebulous goals is probably a feature, not a bug. If your social scene is into sponsoring charity and furthering rational values in a general sort of way, but is intentionally apolitical with regard to religion, politics, or the other major alignments of the time, you're essentially contributing to a partial inoculation against ideology. This strikes me as both prosocial and instrumentally useful to members.

It's common for these societies to require some sort of initiatory ordeal. Anthropologically these are pretty common and serve several purposes, but most importantly they're a hard-to-fake signal of dedication and in some sense competence, functioning as a filter for people with high maturity and executive function and against dilettantes and freeloaders. I think some version of this this has the potential to be very useful for us, albeit with the caveat that it's probably the most cultish practice of the Masons and their various peers. It might not need to take the form of an ordeal, though; Burning Man culture, for example, has similar barriers to entry implicit in the logistics of getting to and surviving in a fantastically hostile desert for a week.

Incidentally, it's somewhat improper to refer to this sort of thing as a cult, as you allude in your links; many such societies have goals and often a body of ritual praxis, but it's rare for them to share any specific orthodoxy, and most are explicitly nondenominational.