As it is relevant & Saves a lot of typing I pinched this from another response.
I know some people have difficulty reading wiring diagrams so I will go through things in a general way.
Power comes to the ignition switch via a fuesed wire from the hot solenoid wire to the B = battery terminal on the starting switch. ( as yours does )
Key turned to Start the switch connects the B & S ( S=start or solenoid if you like ) terminals. ( as yours seems to be doing )
From there the battery voltage goes through some of the safety switches ending up at the trigger wire on the solenoid, where it still should be 12 V ( in relation to earth ).
Starter solenoids come is a variety of types some will switch on the starter motor if they get any voltage at the trigger terminal while others will not trip unless they see more than a specific voltage at the trigger terminal . ( as your does not seem to be doing properly )
This is done to protect the starter motor from low voltage starts which will draw higher amps than the motor can handle.
There are two independent safety circuits.
One prevents the engine from cranking and the other earths out the magneto & prevents the engine from running.
Some of the switches will be in both circuits and some are connected to each other. Some work in a line A to B to C to D etc while others work independently .
If the starter gear "pops up but does not rotate " yes it is cactus.
May be repairable but we don't know which starter it is as we don't know which motor is in your STX38
