My two cents: from the point of view of PDEs I think that a particularly illuminating reference is the recent book of Fiorenza [1]: in this context, the Lipschitz/Hölder continuity requirement is the classical means to control e.g. the behavior of the inner normal vector to a Lyapunov manifold (i.e. a manifold whose representing function is locally Hölder continuous), obtain several classical results on the solvability of boundary value problems for elliptic equations and analyzing the behavior of potentials across a (hyper)surface discontinuity (the classical Plemelj formula is proved assuming Lipschtz continuity of the boundary value of the given harmonic function).
Reference
[1] Renato Fiorenza, Hölder and locally Hölder continuous functions, and open sets of class $C^k$, $C^{k,\lambda}$ (English), Frontiers in Mathematics, Basel: Birkhäuser/Springer (ISBN 978-3-319-47939-2/pbk; 978-3-319-47940-8/ebook), pp. xi+152 (2016), MR3588287, Zbl 1366.26003.