car Ignition switch relay maintain closed after remove power
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This will almost work.  The idea is that the both the motor and the relay is powered as long as the relay is closed.  So we've made a latching relay.  In position 3, everything is powered, and hopefully it will stay that way as the switch changes to position 2.

There's nothing magic about the ignition switch.  Position 0 has nothing connected to it, and position 3 just springs back to 2 when you let go.  Position 1 may be wired such that it is on in position 2, but since we don't use it, it doesn't matter.

The problem is that the relay will drop if it breaks the connection while switching from Start to Run.  If Run stays energized between position 3 and 2, you're golden.  Alternatively, the motor may supply power (as a generator) briefly during the switch return.  A third possibility is to put a capacitor across the relay coil, though a fairly large one would be required.