but what I read about, is that the “sound” is created directly at the neurological level (i. e. directly in the brain) and it does not register if one records the sound.
That is certainly one of the various hypotheses - and it may have some element of truth.
However, it seems people have also been able to record the "sounds" using a variety of microphone types. See this quite detailed paper:
Instrumental recording of electrophonic sounds from Leonid fireballs
I've only skim-read it so far - and I don't know if the researchers had sufficiently electromagnetically screened some of the various microphones so that they were only able to respond to air vibration, and weren't picking up EM waves directly.
If we were natural neurological radio receivers, though, I suspect we would be hearing stuff all the time - from numerous audio frequency EM emitters (such as the electricity grid).
Edit: I've just noticed that the above study was of the 1998 Leonid storm - which was the same one I saw. As I said, I did not hear a thing during that event. However, as I was lying down on the sand in a desert, there was little, if any, vegetation nearby. As I mentioned, some people have claimed that trees (and especially needle-foliage types, like pines) respond to the waves well.