If jumping at the starter works. You can eliminate the starter, cables and battery.
The signal is not getting to the starter. To confirm. Run a test light to the small lug on the starter and see if it lights up when the key is in the crank position.
If this diagnosis proves out. Concentrate your efforts back from that wire. It should be purple and goes up to the bulkhead connection on the firewall. It will only go hot when key is in crank position and shifter switch is in Park and Neutral.