Long story short, there must be one bookable night (for which you are not booked and do not stay at the hotel [under your name at least]) between your checkout and check-in at a single property for it to count as two stays.

Even if you have a 9PM LATE checkout on Tuesday and an 8 AM EARLY check-in on Wednesday, this would still count as two stays since you were not an in-house guest for the hotel books on Tuesday night...