LPL,
There can only be one captain at any one time. Therefore, if your instructor has logged P1, you must log PUT. If you log P1, your instructor can not log the flight. Either of these is valid, but it should be agreed between the two of you before the flight. It would be normal (in my experience) for your instructor to log P1 and you to log PUT, as your instructor said.
What is less clear is whether you can log PUS. There have been several lengthy debates on this subject on this forum in the year or so I've been around. I've read the CAAs documentation, and my conclusion is that it isn't at all clear whether you're allowed to log PUS in this case or not. There's nothing to specifically forbid it. However, the only time PUS is mentioned is in connection to a flight test. A club checkout is not a flight test, so my personal belief is that you're not allowed to log PUS for this flight. This interpretation seems to be shared by most, but not all, PPRuNers. Hence PUT is your only option.
Hope it's clear as mud after that!
FFF
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