I would start off by cleaning the battery connections well- use a file if you need to and tighten them securely with new hardware. Let us know how it goes.

Also check other connections such as engine mounting bolts to the frame.

The fact it doesn't even click the first time indicated the solenoid isn't getting 12V to the small terminal.

Just for "grins", on the first attempt-
Hold the key to the start position and "manipulate" the clutch/brake pedal and "jiggle" the attachment clutch lever.

Agree with both of the above. Jumping with good jumper cables directly from a known good battery to the starter will help determine if the problem is indeed in the wiring, safety switches.

Walt Conner