2002 Cadillac STS starting

I would focus on the starter. On the rear bank of cylinders, near the power steering pump, is a connector that has leads running under the intake. If you disconnect this and apply power to the large purple wire, this is the solenoid signal wire. See if it acts the same. If it does, you have a starter fault. You can try adding some voltage via a jump box or charger just for grins, but 12.2 should turn the motor over.