OSR the movement began as a reaction against 3e mechanics and, to a lesser extent, the trad-style campaign shared by 3e, 2e, and late 1e. This OSR only makes sense in the context of D&D. Other games can be old, other games can be old-school, but only old-school D&D needed any kind of renaissance to pull it out of mothballs and get non-grognards playing it again. (Most other games don't change their mechanics so drastically between editions.) TSR D&D focused forums tended at this early stage to talk about D&D 95%–99% of the time, because that was the focus of the forum; and if other games were brought up, it was only ever because grognards tend to have fond memories of Tunnels & Trolls, The Fantasy Trip, Traveller, Boot Hill, Gamma World, Metamorphosis Alpha, Star Frontiers, Gangbusters, and so on. This was the OSR of Dragonsfoot and the K&K Alehouse.
OSR the play-style developed out of OSR the movement. It came about by examining Gygaxian D&D (AD&D 1e at first, but LBB OD&D and BXcetera OD&D supplanted it as Swords & Wizardry and Labyrinth Lord overtook OSRIC in popularity) and figuring out how the game was "originally" "supposed to" work — a combination of history, historical revisionism, ad hoc solutions, and innovations based on research and new experiences with these rulesets. In other words, the play-style comes from "what makes OD&D sing as a system," so it's deeply rooted in OD&D mechanics; but the principles that come out of that can nevertheless be applied to practically any other game system. This was the OSR of Philotomy's musings, the Finch primer, and blogs.
OSR the creative scene took off once a few things happened. Early on, retro-clones started to get adapted to genres other than dungeon fantasy. (Mutant Future is a good early example.) This, in combination with a play-style emerging that could stand apart from game mechanics, led to the creation of OSR-adjacent games with non-D&D mechanics, trying to either innovate away from D&D or to get at some purer ideal of OSR principles. This gave us "NuSR" and later "FKR" games, often very light, often very focused on that small-scale "OSR at the table" play-style (lethality, rulings-not-rules, heroic-not-superheroic, player agency, etc.) but just as often utterly ignoring large-scale campaign structure (character stables, alignment-as-player-faction, 1:1 STRICT TIME RECORDS, 1:20 ref to player ratio), with open tables and so-called "West Marches" hex crawls and hard-landscape immersive sim living somewhere in the middle. This is the OSR of Google+, the Principia Apocrypha, and Questing Beast.
The OSR begins with D&D, but it ends as a "culture of play" capable of standing alongside trad and storygame.