TFI,
The answer makes itself obvious if you ask "who will own any accident that occurs?".
There can only be one Commander, this must be agreed at the start of a flight and recorded in the techlog. Logically you can't start a flight with one person as the Commander and change this at the end if the check was successful.
Therefore the Commander logs P1 and the other bod logs:
- nothing (Commander not an instructor)
- pu/t (Commander is an instructor and training given)
- p1 or p1/s (Commander is an instructor and interprets rules logically).
...I think I'm agreeing with you.