I had a speedrun rejected, because my timer wasn't on. Of course you can just look at the time below the video (there is also ingame time), to show how long the run took. However my run was rejected nonetheless.

So my question is, are there general rules about this kind of thing (I suspect it happens from time to time), or is this purely handled on game to game basis?

And if there are general rules, where can I view them?

All rules go on a game by game basis, because it always depends on who moderates them in the end. Being rejected for not having a timer on seems a bit shitty, although having an timer does allow you to verify that the video hasn't been sped-up much easier.

Game mods who don't know how to time from a video should step down from moderating.

The time is inherent to the video. If there's not an autosplitter for the game, having the timer on the screen is completely superficial and not meaningful.

A few games have chosen to implement "time without loads" which may require an autosplitter in that specific case.