Many mathematicians prefer to have (at least) two equivalent and elegant(!) but different definitions of the same notion/topic/... (*such definitions either both exist or one proves one of them or even both*).

One of the reasons might be that one definition may look as weak as possible, while the other one may look as strong as possible.

Then when you want to prove that a construction/theory/... is an example/model/... of the given theory then you use the seemingly weak definition.

But when you want to prove hard theorems that follow from the definition then it'd be much easier to apply the strong/advanced definition.

There is intellectual energy stored between the weak and the strong definitions.