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.