TwoSix
Everyone's literal second-favorite poster
I think the sticking point is "the world around them is doing stuff". I think that's really a critical difference between the orientations of play.The players drive the game. If they are passive, literally nothing happens. They sit there and stare at the DM who stares back. The sandbox is over and the DM has to start putting in all kinds of hooks and set them on adventures, etc. Not a bad way to play, but that's not a sandbox. If they are proactive, they set their own goals and desires and start taking action to reach those goals and desires. The DM is entirely reactive to their declarations, even when the world around them is doing stuff.
What the "world around them is doing stuff" means, from my perspective (and from reading various sandbox games oriented books, like Kevin Crawford's X without Number series), is this.
The GM has created a model of a setting. This model is mostly mental, but often supported by physical notes and tools (campaign wikis, encounter tables, gazetteers, maps, etc.). This model is generally focused on important factions, nations, NPCs, etc.
The living world GM is taking this model and then running an objective (although there are differences in opinion in how much objectivity is possible) SimCity/Paradox strategy style set of mental heuristics, possibly supplemented by physical tools, to determine what these various factions and NPCs are doing, how they're interacting, and crucially, what impact the PC actions will have on these interactions and how these interactions will possibly impact the fictional space the PCs currently inhabit.
What this means at the table is that this process will result in new encounters or scenes, and critically these encounters/scenes are NOT generated as a direct result of PC's desires or actions. They can of course be an indirect result, but they are not being generated as directly pursuant/focused on PC goals.
They are the needs of the setting to invoke its own agency on the shared narrative. Because if the setting doesn't have its own demonstrated agency, then how else do you demonstrate at the table that the world is really living and operating under its own heuristics?