Your experience with the community seems similar to mine. The community spirit/integral type is definitely Alpha or at least Fe/Ti, with a lot of activities and conversation about random things. Math club was big, so was hanging out in the common areas, and staying up all night doing problem sets together (because they are so hard this is expected and not considered cheating).
I met a variety of different types, I might as well get this down before I forget completely (it's been 10 years):
Students
LII (me): interested in math as a framework for physics/philosophy and understanding reality on a deeper level. had plans to do a PhD but eventually became disillusioned with academia and unwilling to make the commitment.
LII (REU): quiet guy, maybe average ability/experience but a careful thinker
LII (conference): girl, moderately social and meticulous
LII? maybe IEI? (conference): we were very similar and got along well
SEI? LSE? some Si ego: nice guy, more into applied stuff?
ILE: a friend of mine, interested in math as a puzzle-solving exercise. went on to a PhD in pure math but eventually went into industry
ILE (maybe IEE) (conference): friendly guy, very outgoing, said I "look like a logician". did not pursue a PhD, became a lawyer.
ILE or LII: very enthusiastic upperclassman, always hung out with another likely Alpha (ILE? he wrote an article about his experience
here)
ESE/EIE: similar to the first ILE except more aggressive, both liked playing competitive games a lot. Also did a PhD, not sure where he ended up after that. Impatient and had zero inclination for any kind of deeper thought about math or philosophy. As far as I know he had no interests beyond math and games.
IEI (conference): very odd and amusing guy, would conspicuously take out a textbook and work on it outside, and make strange comments about the nature of reality and society, things like that
ESI (or EII?): seemed really insecure about his math abilities, was a nice guy and friend of friends but I didn't really get to know him, we didn't have much in common.
ESI?/IEI? something ethical Se-valuing: told me that LateX was already good enough and didn't need to be improved despite being around for 20 years

apparently went on to do CS
LIE: seen as a nerdy outcast, didn't really participate in the "fun" social aspects of the community too much. took pure math but (double) majored in actuarial math?
LIE or EIE: similar social pattern to the (first) LIE, very neurotic
LIE (maybe ILE?) (f): again similar, yet more talkative, double majored in physics
a couple very introverted girls: may have been ILI or something, didn't really get to know them but both were actually in high school to begin with
likely ILI: probably the most socially awkward of anyone in the class, didn't really talk at all
ILI (maybe IEI): deliberately antagonistic, would start arguments just to annoy people basically. Funny enough one time he did this to me and started ranting about epicycles or something and then a TA (LIE?) stepped in and pointed out that they were just doing Fourier analysis

EII (TA): similar attitude to me regarding using math for philosophy etc., colder outward appearance; very dedicated to work. his friend group (upperclassmen) seemed somewhat "edgier" and more exclusive.
SLI (possible LSE): really liked the beauty of abstract math but had difficulty doing it himself (his words) - he was very lazy about schoolwork, ended up getting bad grades, but later took some physics courses, proved himself and got into grad school into quantum computing. Much more comfortable in the applied sphere it seems.
LSE (REU): one of the more outgoing people I met, however it seemed like he started a relationship with one of the girls (ESI??) and I didn't talk to him much after that. He was fascinated by the idea of ordinal recursion as an "infinite process."
LSE? (REU): kinda nerdy, seemed talented, I remember she said she didn't like to watch movies because she couldn't sit still for that long just doing nothing
Professors
LSI: probably the best professor I had - very clear explanations, however he did insist on doing things the hard way sometimes inexplicably. Very punishing problem sets. Would make one joke per class to entertain people.
IEE: very unclear explanations by contrast, the students ended up criticizing him a lot. however had an intuitive grasp, preferred geometric visualization of things.
ILE: I later audited another of his courses - good balance of rigor and intuition, a bit of a performer.
ILE?: also a good lecturer, is active on MathOverflow despite being somewhat older
LIE: sort of a loudmouth jerk, there were rumors that he took speed to work more
LSI?: only took his class briefly, sharp guy though. made some interesting comments about how math can never be 100% rigorous.
ILE: kind of a "big fish in a small pond", a bit more dramatic than most ILEs, I also considered EIE
ESI? IEI?: we did not get along, let's put it that way.
Overall I find that all the NTs and introverted Betas are the most natural fits for pure math - most of the other types seem a bit like fish out of water, but some of them make it work.
Famous mathematicians:
Evariste Galois: EIE
Andrew Wiles: LII (has some good interviews on how he thinks about math)
Terence Tao: SLI (maybe LSE)
Ada Lovelace: ILE
Cedric Villani: Beta NF
Rene Descartes: LII
Grigori Perelman: ILI
Alexander Grothendieck: IEI
John Baez: ILE
Isaac Newton: ILI
Paul Erdos: IEE or ILE
Ted Kaczynski: LSI