The two small terminals are 'S' (start, the trigger for the relay, coming from keyed switch)
And 'I' (ignition, the retard trigger of the ignition module, which should be powered while the starter motor turns) your "cylinder by the transmission"
So, really all the owner has done is add a start button.
Taking power from the battery side and bringing it to the trigger point.
Id suggest they have a problem with the neutral safety switch or its associated wiring.
Probably didn't know how to deal with it and added a bypass.
I walked Gary through troubleshooting his clutch safety switch.
He has good pictures.
Do you want to do some diagnosis and repair?
Jim,
Lil'Red is a '87 F250 HD, 4.10's, 1356 4x4, Zf-5, 3G, PMGR, Saginaw PS, desmogged with a Holley 80508 and Performer intake.
Too much other stuff to mention.