|The starter solenoid (a) switches the very high current used to operate the starter motor, and (b) moves the Bendix gear at the rear of the starter motor so that it engages the ring gear on the flywheel.
Once engaged normally, the Bendix gear should disengage from the ring gear by itself only when (a) the engine is turning over faster than the starter motor, or (b) the solenoid is no longer forcing the Bendix gear to engage the ring gear.
Conceivably the starter motor could slow down suddenly for some reason -- jams, perhaps -- leaving the engine momentarily turning faster than the starter, in which case the Bendix gear would disengage.
A very badly worn Bendix gear or ring gear can sometimes cause more or less arbitrary disengagement. This becomes even more likely if the engine should fire a couple of times.
If your ring gear is missing several adjacent teeth, the Bendix gear will disengage. The ring gear can be examined in detail while the starter is dismounted.
Evaluating the Bendix gear requires dismounting the starter (a very simple task in a JH). While some aggressively abraided steel at the engagement points is acceptable, the gear should not show much tooth wear, and the gear should move smoothly back and forth on its shaft when forced by hand or its internal spring. Even if no problems are noted, one should thoroughly clean and relubricate the gear, shaft, and bushing using light motor oil, then reinstall the starter and see if the problem persists.
Since you state that the starter continues to run after the Bendix gear disengages, it isn't too likely that there are problems in the solenoid or its wiring. About the only thing I can think of in this area would be that the solenoid, its wiring, the starter relay, or the ignition switch, might overheat or fail after a few seconds, and that at the same time you have sufficient conductive crud inside the solenoid to allow continued current flow to the starter motor. For the JH, one can disassemble the solenoid for repair (I haven't done that myself, but the task looks easy enough). One would clean the mechanical and electrical bits, polish the electrical contacts and apply a very thin coat of dielectric grease to them, then reassemble and test.
All in all this sounds like a fun problem. I'd be quite interested in hearing what it turns out to be.