This question is a cross post from Math.SE. I have requested the migration of the question, but unfortunately it is not possible after two months of posting. I also have found this related question, but in my opinion it is not a duplicate from mine.
I was reading about geometry in metric spaces from different books, two of them are: (1) A course in metric geometry by Y. Burago, D. Burago and S. Ivanov; and (2) Metric spaces of nonpositive curvature by M. Bridson and A. HÃ¤fliger. Both develop the Alexandrov’s approach to curvature, which uses comparison triangles with the constant curvature model spaces.
For a normed space $ X$ , the following statements are equivalent:
 $ X$ has curvature $ \leq\kappa$ in Alexandrov’s sense, for some real number $ \kappa$ .
 $ X$ has curvature $ \leq 0$ in Alexandrov’s sense.
 The norm on $ X$ is induced by an inner product.
So it seems to me that Alexandrov’s approach is not very informative in the normed case. On the other hand, a geodesic space has nonpositive curvature in the Busemann’s sense if its metric is convex, in general this is a weaker notion than Alexandrov’s, and in the normed case the following statements are equivalent:
 $ X$ has nonpositive curvature in the Busemann’s sense.
 $ X$ is uniquely geodesic, that is, every pair of points is joined by a unique geodesic (the linear segment between them).
 $ X$ is strinctly convex, that is, the ball in $ X$ is strictly convex which means that for every pair of different vectors $ v$ and $ w$ of norm equal to $ 1$ we have that $ tv+(1t)w$ has norm strictly less than $ 1$ for every $ t$ in $ (0,1)$ .
So it seems to me that this weaker notion is the appropriate notion for nonpositive curvature in the normed case and I think also for finsler manifolds. I have never studied finsler geometry, but I am very interested in studying metric geometry from this approach. And I do not know where I should start.
My question is: What is a good introductory book about finsler manifolds from the metric geometry point of view? What is a good introductory book for the Busemann’s approach? If there was not an introductory book available, a reference to an advanced one along with references that cover the necessary background would be very welcome.
In Math.SE, user @HK Lee has suggested the paper On intrinsic geometry of surfaces in normed spaces by D. Burago and S. Ivanov. And I have found the following references, although I need the advice of the experts:

An introductory textbook by A. Papadopoulos about the Busemann’s approach: Metric Spaces, convexity and nonpositive curvature.

A textbook by H. Busemann: The geometry of geodesics

Two interesting papers by H. Busemann: The geometry of finsler spaces and Spaces with nonpositive curvature.
Thanks in advance!