I have converted to 12 volt and I do have the Diagram you attached. Just trying to figure where everything goes. Also does the switch turn off to ?the right or to the left?
If you have the correct housing and ignition turning the key to the right from the center Position should be off and you can remove the key
If you can turn your key either left or right from center position then you probably have an incorrect part.
Probably have the housing for the no-key feature, the no-key switch has a little handel On it
Can you can post a pic of your set up? I have both type housings And can compare them
In any case just wire it per your 12 volt conversion kit