I would say the second starter motor you fitted has a faulty starter motor solenoid & is bridging internally causing the car to turn over as soon as the battery is connected, but the fault is intermittent hence it not doing it now ... Remove that starter motor & discard it ... With your other starter motor the 3 wire one before you refit it hook up jumper leads to it, so you need the earth lead on the Starter body & the positive lead on the solenoid terminal with the nut on it, then get a 4mm wire or so & bridge it from the positive terminal / jumper lead to the blade terminal on the solenoid, hold the starter well & it might initially spark when you connect it but also should throw the pinion out & spin too .. The other short wire you mention should be for a feedback wire possibly to the ignition coil to help starting when cranking ... If the starter doesn't spin when you connect the spade terminal disconnect it & try bridging the short wire, as it could really be either wire that energises the solenoid, if it doesn't spin / work with either wire the starter is faulty ..