When we think of thunderstorm in terms of mythology we usually think of gods being angry, sad or at war. I wonder if there are cultures out there that think of thunderstorm are just Gods farting.
But he is riding his carriage during combat with the giants and then the actual lighting is when he throws his hammer? But I could be wrong, it was a couple of years since I studied this
Thor is not just the god of thunder, but weather in general. Thunder was his carriage and lightning sprang from his eyes. I suppose thunder could also be from his hammer Mjølner.
I've heard your version too, but I think think that's a modern revision, including the lightning from his hammer Mjølner.