I never thought of this as an issue, although clearly it can be. I always thought the rules were crystal clear that there was 8 hours of rest, 1 hour of preparation, and a daily limit on spell slots regardless of how much additional sleep and preparation were carried out during the same day. I still think that, actually, but I am intrigued that there is disagreement about it.
Since the problem seems to be the lack of a definition of the word “day”, I would suggest using any of those available rather than trying to construct one from the rules. I suspect that if the word day was being used to mean something besides what it does generally, there would be a definition. Otherwise, it is not a particularly ambiguous word, and I would think required no more definition than the word dawn when it is used, for example.