The rules for pilots are in 14 CFR 117. Passengers do not have enough information to figure it out themselves.

For F/As, the cutoff time is when the cabin door is closed. For pilots, it is when we begin the takeoff roll.

Some ultra long-haul flights are operated under a fatigue risk management system exemption where additional restrictions are added in order to allow a longer duty time. My fleet/base doesn't have any FRMS flights so I don't know specifics.

Reserve pilots are staged in order to back up specific long-haul flights. That is, the start of their reserve availability period is set so as to have them ready to be assigned to the flight if a delay causes the original crew to time out. They will also have a limited window of available and the delay can stretch to the point that they time out for the flight as well. Some days they may not be available because they were needed for other flights, such as when there are been large scale irregular operations.

Crew swaps are also used by taking a later-reporting crew from a shorter flight and swapping with the crew that timed out for the longer flight who is still legal for the shorter one.