There seems to be plenty of such examples in the theory of Sobolev spaces and their refinements, such as Besov and Triebel-Lizorkin spaces, which have several definitions equivalent in non-obvious ways. One famous documentation of this is the [H=W paper][1] of Meyers and Serrin.


  [1]: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC300210/