Doom and Wolf3d and many other multiplayer games of the 90s (including some I worked on) were deterministic/lockstep and machines only needed to exchange inputs (in a deterministic manner ofc).

Quake was completely different. The client/server term was aimed at describing that the game state is computed on the server, updated based on client inputs send to the server, and then the game state is sent from server to the clients for display. Various optimizations apply.

Deterministic/lockstep games more often used host/guest terminology to indicate that a machine was acting as coordinator/owner of the game, but none of them were serving state to others. This terminology is not strict and anyone could use those terms however they wanted, but it is a good ballpark.

It doesn't need a client server model, but it does need a message pump design.

Then you record the messages as they are recieved, and if networked, tx and rx the messages in the main pump loop.

If not networked, everything still works as normal: game engine itself never knows the difference.