This seems relatively simple, but perhaps I am old school, it used to be defined in the front of CAP 407 (log book), I'm not sure where PICUS, SPIC or PICUS comes from, perhaps euroland.
It is quite simple:
P1 - Pilot in Command (solo or with PAX)
P2 - Co-Pilot
P1 u/s - Pilot in command under supervision (Skill test or renewal). Note as long as P1 does not have to take the controls.
PUT - Pilot in training
SNY - Supernumery ( In this case occupying the jump seat during an observation)
The only oddity regarding the above, is the scenario or let say an A320, where the first officer is PF (pilot flying) and the captain is PNF (pilot not flying), in which case, I would say the captain is P1 and the first officer is P2, as regardless of who is the handling pilot, the captain is the commander and hence P1. I have known some first officers who consider PF as P1 u/s or P1, others may not agree with my interpretation.
The one thing I am not clear about is where an aircraft is deemed single crew, such as a Citation or King Air, yet opts for a second pilot or pilot assistant is that logged as P2 or SNY?