This is a question of terminology (and also relates to philosophy).
Very often we have a certain term, such as "differentiable function", and then we have a definition:
(1) def. A differentiable function is a function such that ...
Then we have a theorem stating
(2) theorem. A function is differentiable iff ...
If this theorem has a correct proof, then we could also have made (2) into a definition, and state that (1) and (2) are "equivalent" definitions.
However, we cannot simply assume that they are equivalent. This has to be proven.
Now my question is one of terminology: Is there an accepted simple term or phrase that captures the fact that "these two definitions are logically equivalent, but they are not semantically the same definition so that their logical equivalence requires a proof"?
For example, I'm thinking of something like "these two semantically independent definitions are equivalent".
A comparison with analytical philosophy might clarify: Frege's concept of "sense" and "reference": it may be that the Evening star and the Morning star are in fact the same particular planet Venus (they have the same reference), but as they are perceived by us, they have different identities (they have different senses). It then requires (physical) proof that they have the same reference. Similarly, our definitions may have the same reference (they refer to the same subset of functions, i.e. are equivalent), but they do not have the same sense (the definitions are different in terms of the meaning they convey when we read them). Hence it requires (mathematical) proof that the two definitions with different 'senses' (meanings), in fact are logically equivalent (have the same 'reference').