When I first drove the Jeep I now have (years before I owned it), you had to turn the key to start *and* press the foot button to start it. The starter at the time was an original core, but had been rewired not to use the starter button on top. Instead, a solenoid had been added. But you still had to manually engage the pinion with the pedal.
A year or two before I bought it, though, the PO had the starter rebuilt (which it sorely needed), and the rebuilder actually knew how a 3A starter was supposed to be wired. When reinstalled, that left the solenoid redundant. When I bought it, the solenoid was bolted to the fender, with no wires going to it, just like yours.
-Paul