The only way this can happen is by still having 6V at the coil after you've turned the ignition switch to off. Unless it is dieseling as Tim has suggested, but you said it still runs without missing a beat, so that does not sound like dieseling to me.
So if it won't start with the ignition switch in the off position then you are still using the foot stomp starter system, right?
This is a 6V system right, not a 12V conversion with the extra wire going from the side terminal on the starter to the coil is it?
With the wiring harness change you talked about I'm guessing you've got the generator supplying voltage to the coil. So even after you turn it "off" and unplug the battery, the generator is providing voltage to the coil.
Get the engine off, disconnect the battery, get out your ohmmeter and check to see if your harness replacement has enabled the generator to supply voltage to the coil.
