Definition of simplicial approximation 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?

Mariano's comment is dead on. It might be helpful to think also of cellular approximation. Given a map of CW complexes $f: X \to Y$, $g: X \to Y$ is a cellular approximation of $f$ if they are homotopic and $g$ is cellular, that is it restricts to a map between the n-skeletons of $X$ and $Y$. I personally don't think a lot about simplicial approximations or cellular approximations because I know they always exist in the settings I care about. The point of having these approximations is then you can make standard types of arguments (like inductive arguments) that are more familiar without losing any generality. 
An interesting question is whether or not cellular/simplicial maps are cofibrant objects in a particular/obvious model category structure on the category of morphisms (if such a model structure exists). Maybe I should ask such a question on some sort of math question site...
