Context-free language and undecidable
Yes, they are the same languages: $L=L'$ (at least the way that I understand the definition of $L$).  We say that the grammar $G_1$ can parse a string $x$ iff $x \in L(G_1)$.  It follows that $L=L'$.  $L$ is defined in a somewhat informal way; good for you for finding a way to provide a more precise formal statement of it.

Indeed, the problem of deciding whether two CFLs have a non-empty intersection is undecidable, so neither $L$ nor $L'$ is decidable.