No..The easiest way is to say, "your leg, you're PIC", under part 1.
As long as that doesn't contradict the name of the person listed on the flight plan as PIC you are correct.
Once again, part 1 says NOTHING about whos name is on the flight plan.
For the third time I agree with this. Who is on the flight plan though defines who is PIC under part 1. Most commercial flights are done on a flight plan. You can't put Tim as the PIC on the flight plan and then jump in the plane and say Bill you will be PIC under part 1 today.