I've noticed that the in-game timer starts at 0:01 instead of 0:00 sometimes when the stage loads

Yes, this is because the IGT carries over from the previous stage. So if e.g. you finish Underground 1 at 16.9 seconds, it'll carryover that high 0.9 to Underground 2.

What about when it happens on Underground 1? Or is it something on my end?

Either from letting the demo play or after a game over

The one I posted previously happened after I did a hard reset twice and started the game normally, here's an unedited footage.

Ah yes, I remember now - it was over 2 years ago that I did some digging. This is what I said in Discord about it:

So basically the IGT is technically running all the way up to pressing start. I think there was some discussion about changing the timing method, however RTA based ones are a bit difficult as speeding up the bonus countdown is very inconsistent, so I don't think any progress has been made on that.

Thank you @BenInSweden for sharing, that is interesting, and @werster for the previous info cuz I easily notice now how the time carries over