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Bill in a sense didn't die, he was erased from the dimension meaning that his existence in that dimension has been removed specifically from that dimension. At best I can say that he returned to his original plain of existence. Stanley losing his memory then regaining it most likely reflects the idea that Bill can return just like Stanley's memories.
A smart guy (whose name I forgot already) once said: "You can kill a man, but you can't kill a thought".
But Bill wasn't a thought, he was a dream demon who existed in the second dimension and then in the demon realm as well as the mindscape of others. And after Stan's mindscape was temporarily obliterated like in an atom bomb, Bill was destroyed/incinerated/evaporated with it, which I think can be considered as dying.