I am asking for a list of concepts which some sources present as definitions whereas other sources pose them as propositions/theorems.
For example, most abstract algebra books will define a group isomorphism to be a bijective group homomorphism. However, after one is introduced to category theory, one realizes that it is an ever-so-slightly non-trivial result that isomorphisms in Grp are precisely the bijective group homomorphisms.
Another example is a $C^k$ differentiable manifold. It is a theorem that every maximal atlas of a $C^k$ differentiable manifold ($k>0$) contains a $C^\infty$ atlas. And thus, the nuances in the terms '$C^k$ differentiable manifold' and 'smooth manifold' are not discussed in some sources.
The final example I'll point out is analytic vs. holomorphic complex functions. I've seen books define holomorphic functions and then say 'analytic' is just a synonym. Whereas I believe that it should be a theorem that every holomorphic function is analytic (where analytic is of course defined to be 'representable by a convergent power series').
The problem with these examples is that without the proper background, I could live my life blissfully ignorant with using these terms as definitions. But I believe that this also robs me of seeing a beautiful result which hints at deep subtleties. So I am asking the community to share their knowledge of other such examples they may have encountered.