My understanding of “statistics” is that it is about either probability distributions, or gleaning some aggregate information from an existing data set which is taken to be either a (not necessarily uniform) random sample from some distribution, or describing the entire population.
Perhaps I am not using the right definition of “statistics”?
Given that your definition of statistics contains nested levels of 'either or', it shouldn't really surprise you that it doesn't really generalize well.
More generally I'd consider statistics to be the applied version of probability theory. Of course in this case the very thing they were trying to compute also fit the definition of a 'statistic'.
If you consider this is to be too broad, then keep in mind that it's simply better when you can apply the concepts and techniques from probability theory to more things.
Perhaps I am not using the right definition of “statistics”?