Pilot Under Supervision, 'wot dat den?
The pilot in the left seat needs an instructor there, the instructor is providing value to the flight - instructors instruct.
It's pilot under training, or PUT, in virtually all cases. Logging P1 is incorrect in virtually any occasion that an instructor is there as you are not genuinely in command - as you've stated, but PUS or PICUS is for successful tests, and pretty much nothing else in single pilot GA.
G