Taking $C^\infty_c(D)$ as the space of test functions seems sensible because it gives you a way to take linear combinations and limits of the $y(\Omega_t)$ in $\mathscr{D}'(D)$. The other way natural way to do this is to choose your $E$ functor $\Omega\mapsto E_\Omega$ to be one for which $E_\Omega = \{f_{|\Omega}: f\in E_D\}$ for all $\Omega\in \mathcal{A}$, at which point you can do the same in $E_D$ (although this latter approach requires you to check that your limits are independent of which extensions you choose - your reference does it like this).