Isn't the solenoid on the Starter? If so, if I unhook the starter and solenoid there is still 12 Volts still registering on the the Hot Wire leading to the starter solenoid without ignition switch on. Got me puzzled. We actually could add a switch in line to the Hot wire to cut the power after it starts but want to find the issue and not rig it.
P.S. - I do have a arc welder (why)? What do you have in mind?