Are we supposed to start the timer from the moment the dropship starts opening or when it is done opening and we can actually move with the character?

The time starts once you can move the character. (on the first frame when standing up)

The time ends once you confirmed the return to orbit. (on the second frame after the dialogue window disappears)

The overlay timer should be started before that (close to the end of the door opening) and should be stopped after return to orbit has been confirmed.

The overlay is there to make calculating the runtime easier but most importantly to make it easier for anyone interested in speed running the game to follow along.

The 1080p rule exists for the same reason. (to make it more entertaining to watch)

So just try your best to have a timer running during the counted portion of the run.

Why is this timed to confirming exit and not to mission completion?

The original release required a return to finish the mission. When the missions were reworked the dropship beacon was introduced, 5 additional inventory spaces and the orbital exchange interface were added. So the timing stayed the same as I didn't see the need to change it and it kept runs comparable.

You can't influence the mission anymore after confirming exit - think of the flight back to space as a long cutscene. From watching a few speedruns in other games, these are generally not counted since it's always a fixed amount added to your time