Incorrect Dave. In the UK (unless conducting training under the integrated system at an ATO) in an SEP, only one crew member can log P1. Therefore, in your checkout scenario, the instructor logs P1 and you log Pu/t, or you log P1 and the instructor logs nothing. The only other exception is, if undertaking an actual skill test or LPC with an Examiner, the Examiner logs P1 and - if successful - you log P1u/s. Since non-examiner instructors are not permitted to carry out such flights, what you're instructor said is wrong.
Completely wrong. A 'checkout' does not exist in regulation but the pilot is P1 being observed to protect the owners asset. An instructor may log P1, even when they are only observing, if as part of their instructor role.