I only ever find the expression "test function" used to denote smooth functions with compact support, which is relevant to distributions. But there are also specific kinds of distributions, applicable to a larger range of functions. So I was wondering if the term "test function" wasn't abstract terminology to denote a type of function that can be tested against a distribution of a given type. So in the context of tempered distributions, a test function would mean a Schwartz funcion (ie $x^\alpha \partial^\beta \varphi(x)$ is bounded for all $\alpha, \beta$), whereas for distributions with compact support, a test function would be any smooth function.
This being true would enable to write properties which hold for various sorts of distributions without having to use "respectively" all over the place.