When is an aircraft considered dispatched?
Technically, before the aircraft is dispatched / released, everything can be inoperative.
However, most everything needs to be in working order to be dispatched / released, with exceptions as per the MEL / CDL.
MEL / CDL procedures are followed until the takeoff is commenced.
If an item is discovered to be missing or inoperative,
A) prior to door close
B) after door closed but prior to block out
C) after block out but prior to flight
there are specific MEL procedures to follow in each of those three scenarios.
Taking the runway and setting takeoff thrust / power is the act of commencing a takeoff.
After that point, equipment failures are treated as an in flight emergency, and flight crew follow QRH / AFM guidance. At that point the MEL no longer applies.
Applicable US FARs for Airline ops:
121.687 Dispatch Release - Flag or Domestic Operations
121.689 Flight Release - Supplemental Operations
135 has a similar regulation.
91k (fractional) does as well.
Plain Jane 91 owners/operators/pilots don't normally operate with an MEL. If they do, they have a letter of authorization with the FAA.