I am thinking one of those 'grounds' is the hold in winding and should get power when the key is on. The other one is the pull in winding. It only gets battery voltage when cranking to retract the solenoid and allow the unit to start.
(No experience on Cadets, but that is how it is on older Grasshopper mowers with Kubota diesel engines in them.) Sounds like the problem is either a bad connection in the wiring or in whatever controls the hold-in circuit. (Grasshopper uses a module. I don't know how Cadets are powered.)
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