The geodesic distance $d(p_1, p_2)$ on a geodesically complete, Riemanian manifold is a metric in the sense of a metric space metric. I'd like to know when other infinitesimal metric structures (e.g. finlser metrics) induce such a global metric and what the exact relation of the two things is.

What is the largest class of Lagrangians $\mathcal{L}: TM \rightarrow \mathbb{R}$ such that the action it assigns to its stationary curves defines a distance function in the sense of metric spaces.

I'm also interested in any procedure that recovers a local metric structure (i.e. Riemanian tensor/Finsler function) from a given global metric $d:M \times M \rightarrow \mathbb{R}$. Any references about this would be helpful.

A few sources state that the Fubini-Study metric on $\mathbb{C}P^N$ can be "easily" recovered from the distance function $d(|\psi \rangle, |\phi\rangle) = \arccos(| \langle \psi |\phi \rangle|)$, what is the procedure for this and are these ideas rigorous?