Yes, yes, yes, we all understand the semantics of Impeachment in the House versus the trial in the Senate. But the fact is, the sole purpose of constitutional impeachment by Congress is to remove civil office holders from the office they hold. There is no other purpose for impeachment under the Constitution.
Is that in the instruction manual ?
Or part of a proclamation from the WeSaySo Corporation ?
The United State's Senate found in 1876 - by a vote of 37 to 29 - that it had the power to try the impeachment of an ex-office holder, no longer holding office.
Furthermore, they voted to convict 35 to 25.
While that latter vote didn't meet the constitutionally-required threshold to sustain a conviction, nonetheless the Senate found as a body (implicitly, by their original vote) that they had a purpose for holding a trial, the consequences of which could have been a conviction.
Otherwise, why bother to hold a trial ?