[folivier]

Another point is when you checked the voltage at the relay or motor did you check with the motor running?
I've seen where a loose or bad connection (as your plug) can show good voltage but when the motor starts the voltage drops. As the motor pulls amps the voltage drops due to the bad connection. Had an A/C receptacle at my son's house do this and found a hot wire broken inside the insulation. The receptacle showed good voltage until you plugged in a fan or vacuum.
Hope your fix did the trick!