The thing to remember is the only legal requirement is Article 28. Para 2(c) requires you to log your operating capacity. It does not give any designations of operating capacity.
LASORS only provides guidance so that hours claimed are countable towards licence issue and subsequent revalidation.
If you are being checked out, there is nothing in law that prohibits one pilot being PIC and the other P1U/S. There is no law that says they must be an instructor or an examiner, or only one pilot may log hours in an aircraft certified for single pilot operation. If you have correctly described your operating capacity in accordance with Art 28 then you are legal. If you are acting as a pilot in an aeroplane then Art 28 requires you to log it. If you have no training or checking role, then you are just a passenger, and can't claim it as time towards qualification; you can log it as PAX or SNY.
If you have no licence but are awaiting it, you can fly under the supervision of a FI, you can fly solo and log P1, you can fly dual and log PUT; you could do a checkout on another aircraft and log it as P1U/S if it was successful.
The instructor can never be a passenger, as you are not qualified to carry passengers. On any instructional flight the instructor can and must log it as PIC.
At the end of the day its your log, you can log anything you like, just beware what you claim using the hours logged. The P3 designation is not used any more (related to 3 pilot operation in the 60/70s).