Since this was a bit long for a comment, I am posting it here as an answer (though it unfortunately does not answer your question per se).

It is not clear to me in general how the topologies are related to each other. In some case one can say something though:

Lemma 5.3. in the book states that a subset is bounded in $L(E, F)$ if and only if it is uniformly bounded on bounded subsets of E, i.e. the inclusion $L(E, F) \rightarrow C^\infty (E, F)$ is initial when we endow $L(E,F)$ with the topology of uniform convergence on bounded sets. Though this does nothing for you to establish that the topologies are the same one can read this as a statement on smoothness into the space $L(E,F)$ with respect to the uniform topology. As the concepts of differentiability discussed in the book do not depend on continuity but only on bornology, the above statement means that the smooth curves into (and thus the smooth mappings defined on) the spaces with both structures coincide. Hence to test the differentiability it makes no difference with respect to which structure you test.
Moreover, you can even get continuity if you assume that $E,F$ are Banach spaces and $M$ is a compact manifold. Then $C^\infty (M,L(E,F))$ is a complete metrisable space and being convenient smooth implies that the mapping is also continuous with respect to the original topologies (which then coinicde with the $c^\infty$-topology of the function space.

Remark: I suspect that for $E,F$ are finite dimensional, the induced topology on $L(E,F)$ should be the same as the operator norm topology. Though I was not able to show it yet, it suffices to prove that the topology turns $L(E,F)$ into a locally convex space (which is finite-dimensional in this case, whence it must be the standard topology by uniqueness of the norm topology)