Starter wiring diagram the truck tries to start without the key?

Seems to be a problem with the starter. The starter solenoid is now in full contact with the battery positive cables causing the starter to crank when the circuit is completed by installing the negative cable.Check if there is any foreign materials lodged between the starter solenoid and battery cable connecting point. Is the starter rebuilt or new?
Did you test the starter before installation? If the Starter cable installation mast is loose, installation can cause the internal contact to go out of position resulting in the shorting. Here is the starter wiring and how to check it.

https://www.2carpros.com/articles/how-to-check-wiring

Check out the diagrams (Below). Let us know what happens and please upload pictures or videos of the problem.

Unplug the starter relay in the under hood fuse box and retry. If problem goes away, the relay is faulty or the starter wire from igntion switch is getting a power supply.

Recheck the starter connections. Remove the starter solenoid wire and retest.

If problrm still exist, check battery to starter cable connection. Ensure it is not touching the solenoid wire. Retest, if problem exist, the fault is with the starter motor.

You said the wires were cut, right? Did you install the wires correctly?

The purple wire (attached to the "S" terminal on the solenoid goes to the ignition key. The black wire (attached to "B" terminal) goes to the battery.