The following book may contain insights of the type you are looking for, at least for the second and third interpretation:
V. Ovsienko, S. Tabachnikov, Projective differential geometry old and new. From the Schwarzian derivative to the cohomology of diffeomorphism groups. Cambridge Tracts in Mathematics, 165. Cambridge University Press, Cambridge, 2005. ISBN: 0-521-83186-5 MR
Although I didn't read the book, earlier articles by the same authors, including a popular exposition in the now defunct "Quant" magazine, were amazingly insightful. Here is how the authors begin their story:
Every working mathematician has encountered the Schwarzian derivative at some point of his education and, most likely, tried to forget this rather scary expression right away. One of the goals of this book is to convince the reader that the Schwarzian derivative is neither complicated nor exotic, in fact, this is a beautiful and natural geometrical object.
I'd like to complement the "third magical power" with the statement that the Schwarzian derivative is a cocycle of the diffeomorphism group of the circle $S^1$ with values in the quadratic differentials on $S^1$, and since the latter space can be identified with the dual space of the vector fields on $S^1,$ it encodes a central extension of $\text{Diff}(S^1);$ the infinitesemial version of this central extension is the Virasoro algebra.