What is “naturally occuring” about vote counts? They are generated by man, using purely deterministic processes...
The sums of the votes are random (in the sense of these kinds of probabilities) - unless the voters in a precinct get together and agree that “176 of us will vote blue and 123 of us will vote red”.
The phrase "naturally-occurring" eliminates man-made numbers like serial numbers, identification numbers, and other numbering systems where systematic alteration of the numbers might be introduced for identification purposes. For example, in a phone number, the first three numbers are a geographical indicator.
Numbers that represent measurements or simple counting are naturally occurring in this definition.