I would think the starter was very unlikely, as if it had stayed jammed on, it would have run the battery down very quickly and probably burnt the cables out after he stalled it. Not only that, but the starter does not turn the engine over that fast, so he would have been able to spot the difference between it trying to turn over, and it carrying on running.
Alex, the ignition key only pulls in the solenoid, if that jammed in, turning the key on and off would make no difference, but as you say, unless by then the starter had already died, it would have tried to re-start.
In my experience of engine run away, the car disappeared in a cloud of smoke, and it was pretty obvious that things were very amiss.
Things to look for are overfilled with oil, or as Rick says, bad turbo, and oil leaking past the bearings, or very worn engine pressurising the sump, and pushing the oil up the breather cct.
Something else, which I do not know in detail, but if there was an absence of smoke, I would be checking the fuel cut off valve, maybe that has jammed.