Sorry I can't always keep up with the conditions vs control logic in these kind of questions so I'll just keep to some general comments.
If you have a condition where the engine spools down below idle with increasing EGT then I suppose you can assume the auto-relight didn't help. So you have to command the fuel off in order to try a relight-restart. The idea of commanding the fuel off is to give the compressors a chance to once again gulp air without being choked by any burner pressure. Kind of like a food processor that has eaten too many tomatoes and is just swirling the soft stuff around at the bottom without being able to eat anymore of the chunks on top.
Of course if you have commanded the fuel off then you have to command it back on if you want to attempt a restart.
Simple flame-outs are just that (EGT is falling) and it's presumed that the compressors are still able to gulp air and what you now need is ignition with a compressor still spooling.
In all this, the key is what is EGT doing since it defines the failure condition, temporary or not.
Somewhere in all this the auto-logic is working but I'll leave that up to others to contribute