I see the problem crop up: a certain mathematical object has many characterizations, any one of which can be taken as the definition. Which do you use when you are introducing the subject?
The first one that comes to mind is the basis of a vector space. Perhaps this is not the best example for the title question of this thread of discussion, but I know that this confuses some students. When I last taught linear algebra, we taught them at least four characterizations. It's not really that any of the characterizations is obscuring or misleading. Rather, each one highlights some important property(-ies). Of course, the better students enjoy seeing all of the characterizations, and they appreciate every one. The less facile students get flustered because they want there to be just One Right Way of thinking about them.
A similar issue arises with the characterizations of an invertible matrix or linear transformation, though at least with a matrix it seems most reasonable to define an invertible matrix as one that has an inverse, namely another matrix that you can multiply it by to get the identity matrix.
The issue comes up in spades when introducing matroids.
