I'm not sure what you mean with "reset the rule." Any trigger will cause your rule actions to run again, but anything that was previously run (or scheduled to run in the future, "wait" events being one exception) will still happen. This is why you need to cancel delays--a new instance of the rule being spun up won't stop anything that's already happening.
If your concern is keeping lights on in the garage when you are there, then you may not need the doors in the rule at all. I assume the motion sensor would see you. But if they don't or you want the lights on sooner and want the doors to have to be closed before the lights turn off, something like this might work:
Triggers: Garage Entry Door open OR Garage Door 1 open OR Garage Door 2 open OR Garage Motion *changed*
Actions:
IF (Garage Motion active OR Garage Entry Door, Garage Door 1, Garage Door 2 any open) THEN
On: Garage Lights
Cancel Delayed Actions
ELSE
Delay: 0:10:00 (cancelable)
Off: Garage Lights
END-IF
If you don't want the doors to have to be closed for motion or door-opening to turn off the lights, you could re-work that a little, maybe:
IF (Garage Motion active OR Garage Entry Door, Garage Door 1, Garage Door 2 any open) THEN
On: Garage Lights
Cancel Delayed Actions
ELSE-IF (Garage Motion inactive AND Garage Entry Door, Garage Door 1, Garage Door 2 any open)
Cancel Delayed Actions
On: Garage Lights
Delay: 0:10:00 (cancelable)
Off: Garage Lights
ELSE
Delay: 0:10:00 (cancelable)
Off: Garage Lights
END-IF
Someone might want to check my logic on that last rule.
I had a hard time figuring out how to account for both cases turning on and off the light under different circumstances. I think that works, though.
NOTE: I am careful to use separate "Contact 1 OR Contact opened" triggers here and not "Contact 1, Contact 2 any open" triggers. This is because a combined "any" trigger will still fire if both doors were opened and one door becomes closed (because one still remains in the open state and that's all it checks after a change). The separate triggers will only fire if any door really becomes open, and that is how these triggers need to be restricted for the actions to work as intended.
Good luck!