Starter

The purple wire at the starter solenoid S terminal should not be hot with key in the run position. If it is, I'd have to back track the circuit. If not, have the starter tested, some of the national brand autostores will test the starter for you. You can do a no load test, take the starter off, lay it in the yard or driveway. Hook jumper cables from battery positive and negative to starter. Negative clamps onto starter housing. Positive clamps onto both the starter solenoid S terminal and the solenoid Bat terminal. See if starter spins and kicks out pinion gear, then when cables are removed, see if pinion gear returns, not kicked out. If pinion gear doesn't return when it should, replace starter assembly.

Some starters may use shims to get the proper relationship towards the flywheel.

I'd have to check feed to and from ignition switch. Maybe the connector has a problem? To go any further, I'd have to look at it. I'd have to test when problem is ongoing.