I think the standard example is [topological dimension][1]:

- Dimension can be defined as either covering dimension, inductive dimension, or simplicial dimension. Then it is a theorem that dimension thus defined gives the same number as the other definitions.

The standard reference for this would probably have been Hurewicz and Wallman's *[Dimension Theory][2]*, published fifty years before Rota's article.

I'd guess that Rota was also thinking of [homology theory][3]:

- One can define homology as singular or simplicial homology. Then it is a theorem that homology thus defined has the same properties as with the other definitions, e.g. the Eilenberg-Steenrod [axioms][4].


  [1]: https://en.wikipedia.org/wiki/Dimension#Topological_spaces
  [2]: https://press.princeton.edu/books/hardcover/9780691653686/dimension-theory-pms-4-volume-4
  [3]: https://en.wikipedia.org/wiki/Homology_(mathematics)#Types_of_homology
  [4]: https://en.wikipedia.org/wiki/Eilenberg%E2%80%93Steenrod_axioms