Thanks GtE, my understanding, too.
As I see it, using plain language 'Pilot In Charge Under Supervision' matches, pretty closely, what's going on with this sort of flying .....
..... right up to the point where the instructor needs to take control.
That's the point where the instructor, most emphatically, must be the one in charge.
Logging PICUS for just the portion of the flight where take over didn't happen would be madness, although a flight where none was required at all corresponds, in some respects, to a test pass.
Thus, outside testing, if an instructor's services are required, Pu/t or Dual is the only sensible choice.
Pretty much what GtE said.
Crucial:
As many have said, getting understandings sorted out before flight is essential.
Happiness resides in common sense, not reliance on regulation.