Interesting.
I was waiting to see if Star would post a wiring diagram for your mower.
According to the one I have found it shows a white wire on the remote solenoid.\If yours has a Kohler it would have a piggy back solenoid.
So this is very confusing and wll become even more confusing now that you have double posted your problem.
I tend to ignore people who do this because I can not be bothered going backwards & forwards between 2 different threads.
If the blue wire is always powered when the key is on then it should not be on the solenoid, it should be connected to the alternator or the fuel solenoid
The only wiring diagram shows a WHITE wire connected to the solenoid.
Originally It seemed to be wired correctly because Kohler usually use a blue wire to trigger the starting solenoid.
\The start circuit goes B > S ( key switch ) C > G ( PTO switch ) through the brake switch white wire to white wire then down to the solenoid.
So you are going to have to trace the mystery wire backwards to see where it goes to.
I rather think it will be the A1 or A 2 terminals on the key switch.