Daniel Moskovich's notion of "support point" to anchor definitions on manifolds is interesting. I suggest a radically non-geometric example: MATROIDS.
A finite matroid is a finite set with a family of subsets satisfying a list of properties. There are different lists of properties. The subsets could be independent sets, bases, circuits, flats, etc.
Given a matroid defined by independent sets (say) there is a canonical way to find a family of subsets that form a basis (say). So the cool thing about this example is that to define a matroid you need a choice of definition.