How to make circuit to reverse dc motor direction with a 12v+ trigger wire
Peter Walker's and AnalogKid's solution to the question, Need a circuit to detect if something has gone from off-on or on-off, is also valid for this question.

The timing diagram is nearly identical.



AnalogKid's timer circuit 'Motor Trigger' is to be used with the value of C3 increased to 100μF for a motor 'on' time of 10s.



The trigger voltage (+12V - engine on, 0V - engine off) is to be tapped from the D+ terminal of the vehicle alternator.



The timer relay 'Motor' is to run the motor for 10s with its direction determined by the trigger relay K1.