Morning! Yes!
If those are backwards that would explain why the voltage was approx. 6 volts on the regulator side and about 10 volts on the coil side when the switch was in the 'run' position.
Assuming the switch works internally, putting the connections right will correct this issue.
ps. It looks like new switches are available, probably can get it locally if your prefer. A new one will should solve the need for the starter button. Because of the way the button is wired to the relay, its possible the relay is bad, so leave the button on until sure the relay is working. Standard Motor Parts US50, or Airtex/Wells 1S6151