I think the standard example is topological dimension:

• 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, published fifty years before Rota's article.

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

• 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.