Hi Max, According to my original schematics, nothing is connected on "RUN". This seems to match well with your re-drawing of it.
I should add that I am not talking about replacing the key entirely. I intended to leave the key stuck in RUN position and splice into the existing wires for the remote starting/stopping.
Also, I would have thought a kill switch is a kill switch and a starter button is a starter button, no matter what the application? One switch has two wires and flicks on/off, whether it be to a coil or to a ground. The button is just a typical momentary push button, again with two wires. A friend has this on a Go-Kart which just has an ordinary engine like on a motor mower. I will have to take a look at how he did it next time I see him. I am having difficulties seeing why this will not work, as it is really just a simple bypass.
Example:
https://www.ebay.com.au/itm/296374168158?Or maybe as I am leaving the key in RUN position, all I really need is the starter button, so something like this, plus a separate on/off flick switch?
https://www.ebay.com.au/itm/354497975400?But basically, both methods are the same.
