I would bet the starter sent back had been in storage for a while, and had built up a film on either the solenoid contacts or on the motor brushes, preventing current from flowing through the motor. The vibration from transport or the turning of the motor shaft would have likely removed the film and thus it passed, or the connector on the car side had corrosion, and the wiping of the contact on the removal and reinsertion cleaned it up enough to get it working again.
If it is a VAG vehicle the ignition switch could also be faulty. There are a lot of recalls for those switches, I just had my one replaced ( as the steering column had to come out to get the clutch pedal assembly out so making it easy) and there have been a few batches with either poor starting issues or catching fire.