My note-taking is based on Apocalypse World and could be summarized as:
Notes are aids in improvisation. They're suggestions (not prescriptions) about what to say so that you create interesting situations in play. Play is pretend, and pretending that the play-world is true is boring to me. Instead, awareness of play as pretend makes games fun (to me and my friends). This approach requires the GM to root for the players while also creating challenging situations. It also helps us resolve play-world consistency problems without notes. These problems are solved in real time creatively. The implication is that notes are sparse enough so that they're actually useful in improvisation (a dozen bullet points helps; an encyclopedia doesn't) but enough to be meaningfully prompted when wondering about what to say next (a blank page is frightening; a dozen bullet points give me direction).
This approach may not work in games where world consistency is required. If you're a player of such a game, what follows may disrupt your game. If you're not, then either a GM or a player could benefit from it.
A fun game doesn't require consistency in the imagined world
The other answers made me think of game approaches that require a world that is consistent. This premise of consistency, as absurd as it may sound in the context of role-playing games, is not at all a requirement for fun.
Different games are fun for different reasons. Some people have fun spending hours crafting dungeons that the players will later interact with. This approach requires players to be keenly aware of the rules that the play-world follows (e.g. travel-time from location to location is irrelevant and sidestepped, or important and played-through; falling off a cliff is routine and feasible, or a grave mistake and a death sentence; NPCs are charitable and forgiving, or distrustful and brutal). Often, a step in the wrong direction could mean death. In a way, this approach assumes the GM is ruthlessly enforcing the game's 'physics engine'. I personally see it as the GM being out to get you. It's not fun for me.
I prefer improvisation, an approach described in the Play Unsafe book. It's more interesting, as it takes wild turns in a world that even I, as a GM, am learning about. There is no physics engine I'm ruthlessly enforcing. Instead, I'm constantly creating interesting situations when wondering what's next. These situations don't need to follow a physics engine. I seek to strike a balance between players getting what they want and creating threatening situations. Sometimes PCs die, but it's not because of a trap-door they weren't able to detect and therefore (physics engine) killed them before they knew it. Instead, players die because of a series of conscious decisions (and potentially bad dice-rolls) that lead them there. Other times, I root for the players and give them exactly what they worked towards.
Each approach's note-taking strategies are different, so my approach is not system-agnostic in that some games stop being those games if 'systematic remembering' (by note-taking) isn't done. But it is system agnostic in that it's a set of principles that could inform your decisions, even if you're playing a more structured game.
TTRPGs are conversations: notes as prescription vs. inspiration
A different approach to 'systematically remembering worlds' is informed by Apocalypse World.
In this approach, you start by clarifying your goals as a player. In the case of my Apocalypse World games, I want to have a conversation that describes cool and dramatic situations (note that all tabletop role-playing games are a conversation, even if it's guided differently in each game). The stuff we, as players (including the GM), say in this conversation are the things that we think are the most interesting to do. Apocalypse World helps us finding those through the Moves the game provides, as well as the GM Agenda, and other tools that could be considered note-taking.
All my notes are limited to the Apocalypse World Threat Map that shows how close or far a threat is from the players (in my case, usually a dozen at a time), and a description of each threat. These less-than-a-sentence descriptions include the threat type, its impulse, its summary and cast, its stake questions, its connections to other threats, and custom Moves or countdowns (again, all of this is specific to this game).
Some of this I fill out while the game is happening (a bit stressful but sometimes necessary). Most is done in between game sessions.
These notes (two or four sheets of paper) are what I have in front of me when playing. I look at them when trying to figure out what to say next. They help me improvise.
And that's it. No need for copious chronologies keeping track of locations, items, NPCs, or whatever. In effect, those four sheets, the character sheets, and the game book are everything I ever need to run the game.
Forgetting isn't a problem when you're aware that you're pretending
What happens if I forget something another player said two sessions ago? What if I forget something I said a session ago? It's not really a problem.
I think notes can have purposes that are different enough to notice. They can remind players as to what the world 'is', or they can give the players things to say to play. What purpose notes serve depends on the game you play (which, again, depends on what you consider fun). My notes rarely describe what the world 'is' beyond the dozen-or-so threats.
In my games, if a player forgets what the world 'is' and says it out loud, the other (remembering) players will usually correct them immediately. If it so happens that everyone forgets a detail, no harm is done. If it isn't a game-changing or logic-breaking detail, nobody cares. If it matters, we make it up again!
It's hard to come up with game-breaking details because the only world that matters is the one we're 'conjuring' right then and there (so details can change) and complexity can be abstracted (so details don't matter). Forgetting an important detail is a puzzle we can solve in seconds through conversation (e.g. "So, wait, is the secret door at your house or at your lab?").
We're not pretending that the world actually exists and therefore demands a systematic physics engine. We're aware that it's a game and so we can play with it (e.g. "Alright, alright, so both your house and the lab are connected by this underground network." I then proceed to think about the threats that loom in those tunnels, and write the notes in the way I previously described).
This improvisational and yet note-guided approach is how my notes are not prescriptive; they suggest interesting things to say that become part of the consensus-built and potentially inconsistent world.
Clarification: my games are consistent and copiousness is compatible with AW
I'd like to clarify two things.
First, this may suggest my games are wacky, with plenty of internal inconsistency. This is not at all the case. In fact they've never been so; we either forget (which I wouldn't be able to tell you about) or find ways of solving consistency problems so that the world effectively has always been consistent (e.g. "Oh, so you didn't use the tunnels until now because you knew a colony of shadows lived there. That makes sense"). The reason I wrote about abstracting complexity is because I think that would be a good solution in case I ever find myself not being able to solve a consistency problem.
Second, I'd like to clarify that it's perfectly possible to play Apocalypse World with copious note-taking (as long as you don't plan scenes, as the rulebook demands). I just happen to play with few notes because I prefer improvisation. Reading Vincent Baker's reflections on the game could be helpful.
