You likely have an EDIC Motor or Fuel Control Relay fault.. you need to get your eyes on your EDIC Rod when you turn off the ignition - is it pushing the Fuel Control Lever on the injection pump FORWARD towards the radiator? If it is being pushed forward and the engine is still not shutting down then its length needs to be increased a tiny bit. The EDIC rod looks like this:
Something like this.. whilst Toyota called it a "Relay".. its got a Printed Circuit Board inside - corrosion often occurs.. I cant be sure where it mounts on the HJ45 specifically, you may have to hunt around for it. My best guess is still the passenger footwell though.
I'm unsure of the pinout on the connection to the EDIC motor so any tests without removing it are difficult to conceive of. If you do go to the effort of removing it then it's possible its the same circuit as for the 47 in which case.. this may be useful.. you're going to need to be pretty competent at some basic electronics to take this diagram, related it the relay and device various tests.. if nothing else you could at least test its supply voltage and earth etc and a basic visual inspection inside the unit is well worth the effort.
That does seem to be doing almost the right things.. the three positions are being set.. its all just looks like its too far towards the firewall.. I would definitely lengthen the EDIC rod and see what happens - there's plenty of room for adjustment looking at the video. I'm pretty confident with a longer rod that it will shut down.