This is going to be a hard one.
The trouble with analyzing Jedi RPers stems from the source material itself: there are about a dozen different iterations of the Jedi Order in canon, both Legends and current, and each one has at least one different interpretation of what it means to be a Jedi. Thus, defining what it means to be a Jedi is extremely difficult for an outsider, and seemingly even more so for the poor souls drawn to writing them.
Nonetheless, I'm compelled by some weird internal masochism to try, so bear with me. As with the Mandalorians, I'm working off my own observations, so if you think I missed something, feel free to chime in below.
I was originally going to try to break this down by era rather than archetypes, but frankly, that's easier said than done, so here goes with archetypes:
The Altruist- One of the more common paths for new Jedi, and probably the most common for new writers. The Altruist writes stories about travelling the galaxy and helping people. Sometimes they're peacekeepers, sometimes they're healers, but whatever the case, they generally try to make the world a better place. The Altruist is the easiest type of Jedi to start, with the possible exception of Padawan, but it's the most difficult to write well. The fact of the matter is, most writers simply don't have the chops to pull off the Altruist for very long and still make the character interesting. That's not an indictment of the skill level of RPers; even professional authors have a miserable track record. If you can find someone who does this well over an extended period of time, you should probably shoot them and have them mounted on your wall, because that's a trophy of the rarest sort.
The Padawan- Another common starting point. The Padawan is new to this whole Force thing. Typically, they'll start off as either young teenagers (small children are really hard to write well), or as adults who discovered their Force sensitivity later in life. The ones who start off as kids tend to be a handful, especially if they're written well, because kids are a handful in general, Light Side be damned. If they're not written well, they're either perfect little puppets or assholes who confuse being maliciously random with being clever. Adult Padawans are often established characters who, for the sake of story or because the author was bored, realized they were Force sensitive. This is much, much easier to write than starting from scratch with a teenager, as there's a lot of character development to be found in trying to reconcile the ideals of the Jedi with a character's worldview. In either case, it's rare for them to stay in the Padawan phase for more than a few months OOC. Those that do tend to have found a niche that they like.
The Redeemed- The third and final of the common starting points. The Redeemed was once a Sith or Dark Jedi or something else Dark Sided that has seen the error of their ways and decided to become a Jedi. Motivations vary, but the common thread is that they find themselves struggling to reconcile their newfound moral code with their old angsty ways. Done well, the Redeemed is an extremely compelling character. The internal conflict is amazing to read as they struggle to master their darker impulses. Done poorly, it's well, a train wreck. They vacillate between holier than thou and barely not a Sith at a moment's notice, and can be insufferably arrogant as well. Bipolar doesn't even begin to describe it. It's like two poorly welded together personalities in the same body, usually welded together by anime hair.
The Redeemer- The Redeemer is the sort of person who cannot help themselves. When they meet a Sith or other Dark Sider, it's not enough to beat them, they have to convert them to the Light. Their pathological desire to convert the Dark to Light is annoying to some, and it has a way of getting them into trouble. Most of them will fight if they're cornered, but combat is definitely a last resort, and they're usually not out to kill. This type played to its logical extreme is fairly rare on sites like Chaos, where Sith have a very large and very hostile presence.
The Old Man- What most types of Jedi tend to turn into if they stick around long enough. With youthful enthusiasm tempered by wisdom and experience, the Old Man is usually seen in the upper echelons of IC faction leadership. Can be further divided into two subcategories: Wise Old Man and Grumpy Old Man. The Wise Old Man draws inspiration from Yoda or Obi-wan circa ANH. Full of wisdom, they can often be found mentoring younger Jedi and perfecting their fortune cookie impression. The Grump Old Man doesn't like you, doesn't like kids, and generally sticks to themselves. That said, underneath all that gruff is a heart of gold. The Old Man is typically the province of more experienced writers, and as such, is a valuable asset to any faction that can snag one. By way of experience, they have a habit of being the best duelist in whatever room they're in at the time, though the really good ones factor age into their performance.
The Young Gun- Younger and more enthusiastic than the Old Man, not all that interested in redeeming or sowing good works, and they don't care much about your ideology. The Young Gun is out to save the galaxy, whether it wants to be saved or not. They're usually the first to advocate aggressive military action against Sith or whatever enemy presents itself at the time. Good examples of the Young Gun are among the easiest Jedi to work with IC, and are generally really cool OOC. They tend to be pragmatic and goal oriented, and they don't often get preachy. Bad examples are in my top three least favorite types of person to RP with. They're overpowered, arrogant, and prone to doing just the dumbest stuff. Worst of all, they'll play the victim quicker than just about anyone else when their actions catch up with them. Avoid at all costs if you value a salt free RP environment.
The Puritan- Takes the Jedi Code very, very seriously. They shun attachments, they shun emotions, and they shun anyone who doesn't. That's not to say that they're all bad people OOC, or that you can't get along with their characters IC. In my experience, the Puritan often knows that their character is a pain IC, and many of them take great pains to make sure that doesn't cause bad feelings OOC. On the flip side, there are the ones who are just straight up dicks. If you're not as hardcore as they are, they want nothing to do with you. They're the most likely out of all the Jedi to hole up in their own little corner and refuse to write with anyone else. More on that later.
The Revolving Door- Could almost be a subcategory of the Redeemed, if there weren't so damn many of them. Though Revolving Door Jedi are still, at least in name Jedi, their alignment changes with their mood, the day of the week, the alignment of the stars, the phase of the mood, and the last time they changed their underwear. They fall to the Dark Side and get redeemed so often, it's not even funny. No one really trusts them, few people can honestly say they like them, and because Revolving Door status is often a result of a lack of experience or creativity, most of them just aren't much fun to be around. That said, when you find one with a bit of self awareness, they can make for delightful characters. One of the best Jedi writers I've ever seen flip flopped between Light and Dark with such panache, their character could walk into any bar in the galaxy and not have to pay for their own drinks. It takes an absurd amount of skill and a great sense of humor to pull that off.
The Grey- Grey Jedi come in two varieties. The first are modeled after Qui-Gon. Though they adhere to the Jedi Code, they don't necessarily follow instructions from any council, instead preferring to follow their own path. These characters tend to be a joy to be around. They're down to earth, good to have in a fight, and often find themselves being the only Jedi in a group of badasses. When they are, they tend to form the emotional and moral center of the group. On the flip side are the Greys who are Jedi in name only. These guys bug me. Done well they can straddle the line between Light and Dark and present a nuanced take on what it means to live in the grey areas that few dare to trod, but I can count on the fingers of one hand the number of times I've seen this done right. Mostly, they just want to think of themselves as good guys will still throwing around Force Lightning and choking people left and right. Most of their posts just leave you shaking your head and wondering why they don't just give it up and go be Sith.
Before we continue, it should be noted that few writers play any one archetype completely straight. Usually, they'll either take an archetype and put their own spin on it, or they'll combine two or more until they get a mix they like.
Now, despite all their differences, most Jedi, and the orders they form, have a few things in common.
First and foremost, they're dedicated to the Light Side. Yes, the Revolving Door types will jump back and forth across the line between Light and Dark according to the day of the week or the cycle of the moon, but even they tend to default back to Light. Though some are more pragmatic than others, most will agree that Darksiders in general and Sith in particular are bad news.
The Jedi orders also share a common tendency to present themselves as the allies of the people. Whether they're helping little old ladies cross the street or protecting a planet from being swallowed up by the Sith, their motives are generally altruistic, even if their actions in pursuit of those motives can be a little extreme.
It should also be noted that Jedi Orders are rarely factions in their own right. Most of the time, they grow organically within an existing Light Side oriented faction. Failing that, a group of writers will get together and attach themselves to a faction that closely matches their moral code.
That's not saying they can't or won't form their own major faction, but there are a few reasons why they rarely do. One of the most important IC reasons is that most Jedi, or at least the ones written properly, are very reluctant to rule. You'll find some exceptions in characters that were born into power, or who had a career in politics before they discovered they were Force sensitive, but for the most part, well written Jedi don't seek power for the sake of power.
Another very important reason is that large groups of Jedi require the cooperation of several smaller groups, and trying to get the smaller groups to work together for any real length of time is extremely difficult. Not impossible, but difficult.
The problem is, writers tend to group up with people they get along with OOC and IC, and that means they tend to form groups with fairly homogenous compositions. Redeemers tend to write with Redeemers, Young Guns with Young Guns, and so on and so forth. While these groups will usually have a mix of new writers from the first three categories, their leadership is often made up of like minded individuals who influence how the rest of the faction behaves.
And since these character traits have a way of influencing IC agendas, what you find is that these smaller groups find it difficult to reconcile their beliefs and agendas with that of other groups. Just about every conclave I've ever seen with the stated purpose of bringing the Jedi together into a single unified front has fallen apart because of seemingly irreconcilable doctrinal differences. They might agree on 90% of what it means to be a Jedi, but that last 10% that's open for interpretation trips them up every time.
There's actually a name for this phenomenon: the narcissism of small differences. Like minded groups with similar goals and ideologies often exaggerate the little differences in between them as a way of maintaining their identity. Sometimes this is fairly benign, but it can lead to huge rifts both OOC and IC.
The most dramatic example I've ever witnessed was back on JvS. Anyone around for this likely knows exactly what I'm talking about, and they're probably rolling their eyes and knocking back a drink, because this is hands down one of the stupidest things any of us have ever witnessed.
There was a group of Puritans based around Yavin, led by a fellow named Dav. They called themselves the Yavinites, but since Dav was the most public figure, everyone else called them the Davinites. For the longest time, they were the single largest Jedi group in the game, but as time went on and the site's activity started to die down, they became increasingly more insular and difficult to RP with. Newcomers who tried to start stories with them were lucky to get a reply to a post within the span of a month, if it happened at all. They straight up refused to participate in events outside their borders, and when they were finally coerced into doing so, they bitched mercilessly the whole time.
One day, a bright-eyed Jedi writer who had been trying to get some traction going outside of Yavin very politely pointed out that their insular nature made it very difficult for anyone to work with them, and that this attitude made writing a Jedi very unappealing to newcomers. They promptly replied by screaming about the savage attack and injustice that had been visited upon them before packing up and mass quitting the site.
Needless to say, this is an extreme example, but nonetheless, let it serve as a warning for anyone tempted to try to unify the Jedi: unless you can find away around the doctrinal problems and OOC gripes, you're fighting an uphill battle. Your best bet is to try to form a loose confederation or mutual defense pact that doesn't rely on anyone compromising for the sake of anyone else.
For outsiders trying to interact with Jedi factions, it's important to do a little research to discover the prevailing attitudes of their leadership. If you're a Sith and you want to get a little hot Jedi on Sith action, most will probably be happy to oblige you. If you're not, well, good luck.
Anyway, if you think I've missed something or have any comments or corrections to make, the comments section is open. Thanks for reading.