is it the starter motor?
The fact that it will turn the car over when it engages suggests that the starter motor itself is actually ok (especially as you don't indicate that it cranks slowly, just "starts fine") . My guess would be that the starter solenoid is sticking and that warming and shaking it a bit (i.e. running the engine) "unsticks" it.
First thing to check though is to ensure that you are seeing volts at the solenoid's switched terminal when the ignition switch is turned to "start". If so, that rules out relay, ignition switch, etc and points the finger of suspicion firmly at the solenoid itself.
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