Your thought process is correct. The starter should run if fuse 9 was hot in the RUN mode. Something is causing trouble in that circuit though and I have another thought. Perhaps there is a ground problem and when the clutch switch is closed it is allowing a voltage to backfeed through the starter cut relay coil. To see if that is so lets try a test. Start the engine and then remove fuse 9. Then with all the other things in place press on the clutch and see if the engine tries to die then. If there is no problem now then clean the grounding connections between the chassis and the battery.