I know the time ends when the saving icon appears at the bottom right corner after leaving the vault, but at a certain point the saving icon just won’t appear when it should and when I either die or reload it doesn’t send me to the correct checkpoint, but to the last checkpoint where the saving icon appears

There's a rare and not well understood bug that seems to happen sometimes after rescuing Vicki Vale that matches what you're describing. That aside, the game should always save when you get there.

In any case, if it doesn't, don't worry about. If you triggered the last cutscene you finished the run, the moderators can infer the proper end time even if the icon doesn't actually come up.