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.
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.