I do not think Hsu's book is a good place to start with, although it has some strong holds-like details in calculation, neither its depth nor its clarity is comparable to
Stroock, Daniel W. An introduction to the analysis of paths on a
Riemannian manifold. No. 74. American Mathematical Soc., 2005.
Don't be deceived by the title of this book, this is by far the most insightful book treating the perspective you stated in OP. Ito and many earlier statisticians want to view the stochastic process geometrically mostly influenced by Y.Sinai and Amari's ground breaking works. But their attention is directed towards J.Doob's gigantic legacy somehow.(This one is irrelevant to OP.)
Doob, Joseph L. Classical potential theory and its probabilistic
counterpart: Advanced problems. Vol. 262. Springer Science & Business
Martingales are treated as a flow of diffeomorphisms, or more specifically a collection of projection mappings. This collection itself has a Lie structure while it acts on the manifold of probability measures naturally. I cannot guess, but I do not think this is an coincidence as the rise of geometric representation theory around 80-90s.
Even for pedagogic purposes or intuition, there is a better alternative to start with, like the classic
Elworthy, Kenneth David. Stochastic differential equations on
manifolds. Vol. 70. Cambridge University Press, 1982.
There is another thing I want to point out, @July said that information geometry is a field of studying geometric side of stochastic analysis, that is slightly deviated from the situation. Information geometry is more directed towards a specific manifold of probability measures instead of studying the (geodesic) flow of diffeomorephisms.