Here's my favorite example. Imre Lakatos' book Proofs and Refutations contains a very long dialogue between a teacher and pupils who debate what are good definitions of polyhedra, with respect to a claimed proof that $V-E+F=2$ is true for polyhedra. It's common that a good definition (or reformulation) of a concept can help yield proofs of theorems, and this book promotes the "dual" view that a proof of a theorem can lead to a good definition in hindsight.
The footnotes of this dialogue show that Lakatos is actually tracing the history of the Euler characteristic in the mathematical literature. In short, both the definition(s) and the proof(s) went through substantial revisions over time.