CK
In the first instance that is an instructional flight. It has one aircraft commander, the PIC is the Instructor, the student is under instruction and logs Pu/t time if on a course of instruction.
In the second example, the requirements for single pilot commercial operations require a working autopilot (with some exemptions for a limited few operators) so can we assume you are talking about commercial ops.
If that is the case the aircraft, say a piston twin, has to be certified by the CAA as a two crew operation and there will be an ops manual stating that and laying down each pilots role. To add to that both pilots will then need fire, first aid, dangerous goods, CRM courses completed and be base checked and line trained on the relevant aircraft.
I was lucky to log some P2 time a few years ago in a PA31 which carried out ILS calibration work, but as far as I know it was the only one in the country approved for 2 crew operation and then only whilst on calibration work.