When I was teaching elementary geometry in high school in Holland 10 years ago, the concept of equivalent definitions was actually part of the curriculum. There the examples were more down to earth: you can either define a parallelogram as a quadrangle in which opposite sides have the same length and prove as a theorem that they are parallel, or as a quadrangle in which opposite sides are parallel and prove as a theorem that they are also of the same length.

Then you can also take the third standpoint of 'pick whichever definition you like best but justify this by proving as a theorem that both definitions are equivalent'.

Actually the third standpoint was kind of like the official school standpoint and we spent a lot of time trying to convey to the students that proving this latter theorem (of both definitions being equivalent) amounts to proving both the theorems mentioned above it.

I remember students found this 'equivalent definitions' stuff exceedingly confusing. This might however be partly be due to the fact that this short block of elementary geometry was the only time they had to work with definitions and proofs at all.