Awyren - as Stevelup states there is no such thing as a 'co-pilot' in a SEP certified for single pilot operations.
I think the confusion in your mind may have arisen regarding the term 'check out' and a 'flight test' for the grant, revalidation or renewal of a rating forming a part of your licence.
If the flight test is carried out by an examiner for the grant, revalidation or renewal of a licence or rating then the examiner is P1/PIC. As the candidate on test you book P1 Under Supervision (P1CUS) for a successful test and Pilot Under Training (PU/T) for an unsuccessful test.
However, if you have a valid SEP rating and you are being 'checked out' to fly another type of SEP, then there is no need for an instructor at all - the flight could be carried out by another member of the Group, for example. This is because, technically, your rating covers the aircraft type. In that case you could indeed book P1 and the other pilot, whether instructor and/or Group member is just along for the ride so to speak (although naturally passing on useful information and checking that you are, in their opinion, safe to operate the aeroplane).
Nevertheless, if you are covering 'differences' training with an instructor on a more complex type, then the instructor will be P1 and you PU/T. The grey area in your mind may be that 'differences' training is defined (tail wheel, for example) and not 'different' type (where an instructor may be the other pilot, but not required to 'instruct').
The bottom line is: are you licensed and rated to carry out the flight unaided (that is, solo)? If you are then you can be P1, if not then by definition it must be some form of training or test and you can only book PU/T or P1CUS depending on the result of that training or test.
Separately, in the commercial world where there are co-pilots (first officers) then a sector where it is the FOs 'leg' (handling pilot) it is booked as P1CUS and where the Captain is the handling pilot (FOs non-handling) then the FO books P2. The FO can never book P1C and the Captain is usually P1. Awyren your third paragraph describes this arrangement, which is meant to apply to airline flying and not the realm of the PPL.
Bose-X makes a good point in that irrespective of the rights and wrongs instructors do tend to want to claim P1 hours for their own purposes. This may or may not be required. If you go to a flying school (ATO) you will probably be told to book 'PU/T' (there can't be two booking 'P1'), whereas with Group flying you will most likely be able to book 'P1' for the reasons described above. That has always been my experience.
Last edited by SpannerInTheWerks; 20th Sep 2016 at 11:38.