My impression is that pre-OD&D is the goal for FKR style play; I've heard the term "Arnesonian gaming" elsewhere. Incidentally, this view would place it in specific relation to the history of dnd more so than in relation to more recent non-dnd-like rpgs. Further, my impression is that in dnd-type games there is a particular relationship between system and setting that they are trying to invert. So, rather than playing a game of 1e, or 3e, or whatever in the setting of Greyhawk, you are "playing Greyhawk." For example, see some of the quotes included in this post:I think I would find it helpful if someone who thinks they "get" FKR better than me would identify some of the rules-heavy systems that are the objects of its critique. As I've said, 3E D&D seems to me to be the core of it; but are there other systems that are also being had in mind?

Play worlds, not rules: Juggling ideas for stone age rpg sessions
… and I don’t mean The Flintstones here. What really is driving my interest these days (in terms of roleplaying games) is to go as far back as possible with the rules. And since it̵…

- “Doing it by the book” was impossible; the book – and the game rules – hadn’t been written yet. (1)
- “I mentioned that I’ve never really ‘played D&D’; I’ve played “something called Blackmoor with Dave, something called Greyhawk with Gary, and something called Tekumel with Phil” (2)
- “We played using whatever tools we needed at that point in the campaign – RPGs, Braunsteins, miniatures, boardgames, poker, you name it.” (12)
- “There’s a lot of nonsense about the way Dave played and organized Blackmoor floating about; a lot of people are assuming that he was working to A Great Master Plan when he wasn’t. He loved to simply play, and he whipped up the game mechanics and ‘history’ / ‘timeline’ to suit the game in progress. I guess that the best way to ‘play like Dave’ is to not over-think the thing – don’t worry about how it all has to make sense somehow.
There is something resonant here even when looking at the rules-lite games of the OSR. On osr forums, there is a lot of talk of "which system" to use. e.g. quesitons like, "do I use basic fantasy or OSE or the black hack?" Sometimes there is an attempt to "match" system and setting, e.g. "what system is best for barrowmaze/stonehell/ultraviolet grasslands?" or "how do I convert Against the Giants for Into the Odd." All those games might be rules lite, but there is a certain obsession over which of those variations of dnd are 'best.'
Part of my particular circumstance is that I have players who don't bother to read rules or principles anyway, so I'm basically running all the 'game' elements on my end anyway. I played a 5e campaign this way, which was done by the book, which for some of the players amounted to them rolling a die and me doing all the math and narrating the outcome. So rather than focus on system, what I want is for my players to turn to a random page of the Ultraviolet Grasslands setting book, look at one of the beautiful illustrations, and say 'this, whatever this is, I want to play this.'