I used to think every map that is fiber preserving is a bundle map in the sense of the definition below:
Definition Let $E\xrightarrow{\pi}M$ and $F\xrightarrow{\rho}N\ $ be vector bundles and $g:M \rightarrow N\ $ a diffeomorphism between $M$ and $N$. Then, we define that $f:E\rightarrow F\ $ is a bundle map that preserve fibers if: $$ g\circ \pi = \rho \circ f$$
But in page 209 at Foundations of Mechanics a map $f:E \rightarrow F\ $ between two vector bundles such that $f\ $ is fiber preserving is defined, but there's a remark saying that $f$ is not necessarily a vector bundle mapping.
Question How could a map be fiber preserving without being a bundle mapping?
A vector bundle mapping has to be a linear map on the fibers (the fibers are vector spaces and it has to be a linear map in the sense of vector spaces). So for a counterexample, you could take $\mathbb{R}$ thought of as the trivial bundle over a point, and any smooth map from $\mathbb{R}$ to itself that is not linear (e.g. $x \mapsto x^2$).