Does the crew rest period start with the completion of the last CAP activity OR does start when the member actually arrives at their place of rest (home, hotel, other).

Crew rest starts at the end of the last CAP activity.

The USAF does a good job of defining Crew Rest and Flight (Crew) Duty Period (FDP). Please refer to AFI 11-202V3 below.

AFI 11-202V3 dated 22 OCTOBER 2010

9.4.5. Crew Rest Period. The crew rest period is normally a minimum 12-hour non-duty period before the FDP begins. Its purpose is to ensure the aircrew member is adequately rested before performing flight or flight related duties. Crew rest is free time, and includes time for meals, transportation, and rest. Rest is defined as a condition that allows an individual the opportunity to sleep.

9.4.6. Flight Duty Period. FDP begins when an aircrew member reports for a mission, briefing, or other official duty and ends when engines are shut down at the end of the mission, mission leg, or a series of missions. FDP for UAS ends at final engine shut down, final in-flight handover briefing, or final crew swap, whichever occurs last.