What is contextual equivalence ignoring non-termination called?
(This is an extended comment).

I may be misreading your definitions, but it seems to me that the relation you introduce, let us call it $\simeq$, is not an equivalence relation because it is not transitive.

If $V_1,V_2$ are two distinct closed normal forms (values), then obviously $V_1\not\simeq V_2$.

On the other hand, the genericity lemma (Proposition 14.3.24 in Barendregt's $\lambda$-calculus book) says that given an unsolvable term, say $\Omega$, for any context $C$, either $C[\Omega]\!\Uparrow$ or $C[M]$ has the same normal form (a value, if we assume to work with closed terms) for every term $M$ (including $\Omega$). This shows in particular that $\Omega\simeq V_1$ and $\Omega\simeq V_2$, hence the failure of transitivity.