The class $V$ of all sets is not a model of $ZFC$, because it is a proper class, not a set.
A model of $ZFC$ is a set (or small class) $U\in V$ which satisfies all the axioms of $ZFC$ when these axioms are restricted or relativized to $U$, even though $U$ does not include all the sets of the $ZFC$ universe.
If a model $U$ of $ZFC$ exists within the universe $V$ of $ZFC$, then its cardinality is "inaccessible" with respect to the universe $V$. Conversely, if an inaccessible cardinal exists in the universe $V$ of $ZFC$, then a small class, or set $U\in V$ exists, which is a model of $ZFC$.
The existence of a model $U$ within the universe $V$ of $ZFC$ implies that $ZFC$ is consistent. $ZFC$ would be inconsistent if its axioms could prove the existence of a model of itself within itself.
I fear that the meta-theorems, meta-theory, and other meta-language are unnecesssary and ill-defined, unless in the universe $V$ of $ZFC$ we are speaking of the properties of a possible model $U, U\in V, U\subsetneqq V$ where the original axioms of $ZFC$ have been restricted to a $U$.