As the king of the OSR I would be remiss not to mention the Free Kriegsspiel Revolution (hereafter FKR), at some point because everybody seems to have decided to talk about it for a minute.
FKR is basically just cowboys and indians with a moderator. There potentially is a character/trait system but usually no numbers or ratings for anything. As there isn't much to these games (this blog post will be longer than many of them), it'd probably be easier to show than tell:
The last paragraph kinda shows how you can connect a game setting to it and bizarrely I don't hate it entirely. Now FKR does blur the line between group storytelling and take several leaps and bounds away from RPGs as we know them. It's also not OSR in the slightest, because the mechanical baseline that everyone at the table sits down with is what constitutes a game. And I don't hate it for that, it requires a shift in mindset and perspective to run and enjoy. Nothing is overly random during character creation but then nothing is completely out of bounds either. GM's steady hand is important in FKR games like no other. There is no real designer to hold your hand in balanced matters.
Difference between FKR and Minimalist games: Into the Odd, Electric Bastionland, and Troika are FKR games and don't know it (or mostly don't). Other OSR games don't fit, including minimalist games like Microlite20 and White Box, because of their incessant need of a mechanical baseline for everything. Searchers of the Unknown is on the edge.
I'm writing my own which is an interesting prospect. This brand of minimalism requires limiting your design to the foundation for play. The difference between OSR and FKR is there is no rule 0 because there doesn't need to be. The benefit being that essentially playing RAW is mandatory.
Now I'm sure there are FKRers who don't feel that way, but when your game takes up 1-4 pages, any derivation from this constitutes playing an entirely different game.
The benefit of FKR is also it's limitation. I've been considering diegetic rewards for a long time. the fact is if you're playing using escalating levels and numbers, while they can be satisfying, they also tend to grow a character in ways not expected or really warranted. Some think the fix for this is having skill levels rather than character levels, but that just shifts the goal posts. People don't always get better at things over time, and if you're taking the step to attempt to remove gamification regarding character levels, there's very little reason not to take the next step and simply remove the gamification entirely.
While things in D&D often get overlooked like material components for spells, rations, water, light sources, etc, these elements would be the bread and butter of FKR dungeoncrawling. Along with armor and weapon breakage which is flatly never considered in D&D. A referee who's mind is free of mechanics can actually focus on these elements better than one looking into action economy and other stupid gamey bullshit. Even OSR has modifiers and stuff that essentially don't need to exist if you judge things on a case-by-case basis.
So FKR is worth looking into but at the same time it's kinda not, as it requires ultimate player buy in to a DM. Players usually use character creation and silly character options as a means of generating interest in a game beyond what actually happens at a game. This is true for OSR and even more true for WotC Editionists.