I have been given the following definition of simplicial approximation in lectures:
Let $K, L$ be simplicial complexes and $f : |K| \to |L|$ be a continuous map of their polyhedra. A simplicial approximation of $f$ is a map $g$ of vertices of $K$ to vertices of $L$ such that $f(\mathrm{st}_K(v)) \subseteq \mathrm{st}_L(g(v))$ for each vertex $v$ in $K$.
However, elsewhere I find the definition that a simplicial approximation is any simplicial map which is homotopic to the original map. This seems, to me, to be considerably more general than the definition I have, since, for example, if $f$ is homotopic to a constant map, then I can have very trivial simplicial approximations in this sense. So my question is, which is the more common / "morally" correct / better definition?