[greglepore]

OK, so the contactor has 4 terminals, which means its breaking both hot and ground (assuming-some just break hot side so 2 in/2 out).
I suspect that with one terminal disengaged the induction of the motor being on the circuit is allowing you to see 120v on the "dead" leg of the 240v circuit (again, an assumption as I'm not an engineer-but Choo Choo Charlie was an engineer).
Contactors are cheap, suspect your issue is there as one leg shows no continuity. Sometimes they weld themselves together in failure mode and when you cycled the coil current it shook loose.
Or the motor is bad and burned out the contactor.
But I'd start with the contactor if you're going to continue to diy this.
Not an electrician, but I stayed at a super 8 motel once.