Impeachment does not remove. Conviction removes.
The House impeaches, the Senate convicts.
Impeachment is akin to indictment.
Senate convicts is actually more akin to Senate removes by presenting facts for a vote in which Senators act as judge, jury and executioner; executioner in the sense of removing from office but not for imprisonment.
Once removed by the Senate, the judicial branch may try to convict in a court and this can result in imprisonment.
I understand what you are saying and it is correct. I just thought it didn’t need to be pointed out that Conviction is necessary.
As for the rest ... The house Impeaches, the trial is held in the Senate and the SCJOTUS presides over the trial.