Get a new solenoid and install back the started with proper insulation.
I currently do not know whether the solenoid is shot or not. The short was on the outside. The inside may be fine.
From the pics it isn’t clear to me that there was a short from the starter motor stud (which looks clean) to something. And that assumes the stud is always energised, not just during starting.
The stud is energised only during start. There was a nut on the stud, so the stud itself was probably not in contact with the carbon.
It turns out that it is possible to rotate the starter by increments of 20º wIthin the baffling. Unless the solenoid would then interfere with something else, that would have allowed ample clearance between the solenoid and the carbon baffling.
Aviathor wrote:
It turns out that it is possible to rotate the starter by increments of 20º wIthin the baffling. Unless the solenoid would then interfere with something else, that would have allowed ample clearance between the solenoid and the carbon baffling.
Plenty of space if you just clock the solenoid 20° CCW . Interesting the installer was not aware of that option.
The solenoid terminal stud in the picture looks blackened. Is it possible that it was this that made contact with the carbon, energising the starter with the engine running? I don’t know if the solenoid is pull-down or pull-up, but vehicle designers have a disconcerting love of pull-down accessories.
Lest you think you’d notice an engaged starter, my Defender drove 1.5 miles with the starter engaged due to a faulty ignition switch with no unusual noises (by Land-Rover standards!) and it only became apparent what had happened after stopping to check the trailer and seeing the vehicle enveloped in smoke!
I have a starter energized annunciator. Never lit.
Our starter energised annunciator was not connected after starter repair. No-one noticed it for about 2 years. No-one looked at it while the starter was engaged. And when we checked after starting, it was not lit. 
Only when the taped-up wire came loose, and intermittently shorted did we find out. Surprisingly each short caused the oil pressure reading to drop.
