Topologically, is there a definition of differentiability that is dependent on the underlying topology, similar to continuity?

Take a look also at No topologies characterize differentiability as continuity by Geroch, Kronheimer and McCarty (1976).

The notion of differentiability is not involving only a topology but a normed space. And in $\mathbb R^m$ all the norms are equivalent.

Therefore the topology used in the definition of differentiability is the one induced by the norm. This topology is unique in $\mathbb R^n$. This maybe different in infinite dimensional Banach spaces.

Finally, there is only one relative topology on a subset $S \subseteq \mathbb R^m$ induced by the « normed topology » of $\mathbb R^m$.