With the way that the day to day operations of private GA is usually conducted it is not surprising that there is confusion. If you are the pilot in command then yes you log PIC or if you wish P1, it doesn't matter which you chose because it is only semantics, P1 uses less ink. You are only a pilot under training where the law requires this.
On any skill test for the issue/renewal of a licence or rating then you are demonstrating your ability to be the PIC for the particular circumstances, but you are being supervised by the examiner. So, you log PI supervised P1/s or PICs as you prefer. The examiner may direct the flight, take control or terminate the test at any point so they are ultimately in command. If you fail the test then you have not succeeded in being the pilot in command in the circumstances so PUT. For all simple types the examiner must always be an instructor also allowing you to log PUT. There is only a requirement to log in a particular way when operating in accordance with the requirements of the regulations.
Conversion instruction onto simple single engine piston types, by example, are not required by regulation, in most cases, although most sensible people considerer it to be wise to do. Such conversions may be part of a group ownership agreement and given by a group member. The person selling the aeroplane may offer this as part of the sale deal. There are many single seat types on which dual instruction cannot take place. None of these examples as with many others require the involvement of an instructor. Club checks such as are routine 'currency checks are stipulated by the club but not by the law . They will usually nominate an instructor to supervise the conversion or the club currency check. Young instructors in particular often direct the pilot to log PUT so they can log themselves P1, this being part of their hour building needs. For me, the pilot should log P1/PIC on these flights. There is no pass or fail or required training to log. It is a deal that you make with the owner prior to whether the owner will let you fly their aeroplane.
So it is not the case that if an instructor is in the aeroplane you must log PUT.