A lot of water is being muddied here because the military term P1 and the civilian term PIC are being interchanged.
UK military log books have a First Pilot (P1) column and a Captain column. So, on a big grey jet, the guy in the LH seat logs P1 but the captain could be in the RH seat (also logging P1 but logging captain as well) or (shock, horror) could be a navigator sitting down the back. The military also allows 2 pilots flying a trainer/2-sticker to both log P1 if suitably qualified but only one logs Captain.
A civilian log book has one, Pilot in Command (PIC), column. Thus it is equivalent to the military Captain column.
In the civilian single-pilot world the only guy who can log PIC is the pilot in command - the captain. Anyone in the other seat of a single-pilot aircraft is either receiving instruction from the captain (who therefore must be an FI, IRI, TRI or CRI) and so logs PU/T, or is just a passenger and can't log anything. The only 2 exceptions are: a pilot undergoing a successful JAA Skill Test or Proficiency Check and flying with an examiner (ie FE/PPL, FE/CPL, CRE etc) and, for guys on integrated CPL or ATPL courses only, a student flying with an instructor as IF safety pilot (this is known as SPIC - student pilot in command).
Using a civilian logbook, club checks with an instructor are PIC for the FI and PU/T for the checkee. Club checks with anyone else are PIC for one of the pilots and nothing for the other.
Solution: scrounge a military logbook!