My question is closely related to this:
On the definition of a geodesic in a metric space
I don't understand why in the definition of the geodesic there is the requirement of constant speed. As far as I understand, the constant speed is given by a particular parametrization. However, the length is independent of the parametrization. Intuitively, the concept of a geodesic is related to the length of the path, so it should be independent of the parametrization as well. Why isn't the following a good definition of a geodesic:
A path $\gamma: I\to X$ is a geodesic, if for any $t\in I$, there exists a nbhd $U$ of $t$ such that for any $t_1, t_2\in U$ one has $$ d(\gamma(t_1),\gamma(t_2))=L(\gamma|_{[t_1,t_2]}) $$
where $L(\gamma|_{[t_1,t_2]})$ is the length of the path $\gamma$ restricted to $[t_1,t_2]$.
This definition includes the constant speed geodesic, but not only. There is something I misunderstand, what am I missing here?