Earlier quoted context omitted.
It wasn't really that much to do with determinism. Quake uses a client-server network model all the time, even when you're only playing a local single-player game. What the demo recording system does is capture all of the network packets that are being sent from the server to the client. When playing back a demo, all the game has to do is run a client and replay the packets that it originally received from the server…
I don't see why it makes a difference for this purpose that you're replaying network packets or controller inputs or any other interface to the game engine. The important thing is that there is some well-defined interface. I guess designing for networked multiplayer does probably necessitate that, but if the engine isn't deterministic it still isn't going to work. There was a twitter thread years ago (which appears t…
The example you mention of demo playback de-syncing when the circumstances slightly change, that is exactly what you get when you only record inputs from the player. Doom actually did this too for its networking model and demo playback system. That relies much more on the engine being deterministic and the runtime environment behaving consistently, because each client that replays those inputs has to run the exact same game simulation, in order for the resulting game states to match.