Trump was President when he was impeached. We are in uncharted territory because there is no precident for someone impeached while in office and then the trial was held while they were out of office.
It will be really hard for them to split the baby, claiming “in office” rules apply, and then also claiming “out of office” rules apply. Which is it?
If the President's term is expired, then the impeachment becomes moot and the proceedings should be terminated.
I would consider it to be like a pocket veto when the Congress has adjourned sine die (at the end of its term). In this case, it's the end of the President's term that vetoes the impeachment.
-PJ