2001 Oldsmobile Intrigue Starter

Check the purple wire at the starter solenoid S terminal, engine running, use a voltage tester, is the wire hot? If it isn't, I'd suspect an issue with the starter itself. Some of the national brand autostores will check the starter for you. Take it off and take it to them. The ignition switch should be spring loaded in the crank position, when the vehicle starts and you release the key, it goes back to the run position. The purple wire at the starter solenoid should not be hot, with key in the run position.