I agree with most of the above. The only proviso is until you have the licence you cannot take passengers, even if that passenger is a pilot or instructor.
So an instructor can authorise the flight as a student supervised solo, or can be on board in the role of instructor. So PIC in one case or Pu/t in the other.
It does make sense to me that if an instructor is there purely in a supervisory role (as in a successful club check flight) with no intention to instruct, then it should be P1S for the pilot and SNY for the instructor. If the instructor intervenes then it would become Pu/t.
This option though doesn't appear to be sanctioned in the JAA definition.