Which is the "proper" definition of a geodesic curve?

I don't really see any advantage to restricting the definition of geodesic to be minimal -- after all, those are just what we call "minimal geodesics"! As you do more geometry (Riemannian and otherwise), you'll encounter many other definitions that are given via differential equations. These all have their local theories -- in this case, we find that every point on a Riemannian manifold has a neighborhood where minimizing geodesics are unique -- and this does not detect the global behavior. But this can be a good thing, because once you've nailed down the local picture then you have firmer footing to ask global questions. Here, we might ask: When exactly does a geodesic stop being a minimizing geodesic?

These words may not mean anything, and they don't really need to, but a similar differential-geometric example that might shed light by analogy is Darboux's theorem, which says that all symplectic manifolds of the same dimension are locally symplectomorphic. That is, as far as the stuff we care about is concerned (namely the "symplectic structure"), neighborhoods of any two points on any two equidimensional symplectic manifolds are indistinguishable. This is true too of smooth manifolds, but not of Riemannian manifolds, since curvature gives us a local invariant with which to distinguish them. (In fact curvature is the only local invariant! But that's another story.) But nevertheless people call themselves symplectic geometers, and indeed there are some very deep global questions in symplectic geometry.

The moral of the story is that in geometry one often starts with an idea (e.g. "the shortest path between two points"), examines the local behavior, and then works to understand how the local story pieces together to form a global picture. This makes it very natural to begin with local definitions such as the one you mention.

A short answer is that the definition via geodesic curvature is much easier to verify, hence much easier to prove things about.

One example of a geodesic being the longest path in some sense is in Einstein's relativity. It is related to the Twin Paradox, where two twins set off from some point in spacetime and then meet again at another point in spacetime, to discover one has aged more than the other.

The geodesic is the path which takes the longest as measured by a clock passing along it. For Special Relativity, this is a straight line and a constant speed, i.e. inertial motion, and any clock going by any other path will measure less time.