From the looks of it, I would say the small rectangular chip-set with five metal contacts at the base and a nut at the top is an Integrated Micro Switch responsible for triggering the starter motor. You can verify this by checking continuity and voltage outputs.
Care of James from https:
Here is the data sheet on the Micro Switch:
The prongs at the bottom in your picture are 1 - 5 (left to right), where 1 and 5 are (+) outputs to your load (the starter), 3 is (+) from the battery, 2 is your switch (triggered by grounding circuit) and 4 is used for diagnostic feedback (unless you are knowledgeable about proportional currents to loads you can leave this alone for now).
Using a basic Multimeter or even a test light, you can check the prongs for voltage. Unplug the starter from the power box and connect the battery to power box. Connect the ground for the tester to the (-) terminal on the battery and check for voltage at prong 3 first (middle). Should be equal to whatever battery voltage is (12V +). Next check voltage at prongs 1 and 5 (far left & far right). Should be zero volts with starter trigger NOT pressed and (12V + ish) with starter trigger pressed. If it has voltage and starter trigger is not pressed, the micro switch may be bad.
One more step to verify. You will need to disconnect the (+) lead on the battery and use a multimeter or Ohms meter to measure continuity from prong 2 (second from left) to the negative terminal at the battery. There should be no continuity when the starter button is NOT pushed and continuity when the starter button is pushed. If there is continuity without pushing he starter button, you have a bad starter button or short to ground somewhere in that circuit.
Let me know what you find.