This is a follow-up to a previous question of mine.
If $X$ is an $R_0$ space (see the linked question), then the quotient obtained by identifying two points if their closures are equal is $T_1$ (in fact, it is the universal $T_1$ quotient of $X$, and this property is equivalent to being $R_0$).
Let us say that $X$ is an $R_1$ space if this quotient is actually Hausdorff.
For example, any topological group is an $R_1$-space.
Does this $R_1$ property have a standard name?
In fact $R_1$, or preregular, is the standard name it seems: wikipedia, it explicitly mentions that a space is $R_1$ iff its Kolmogorov quotient (identifying topologically indistinguishable points) is $T_2$. So a space is $R_i$ iff its Kolmogorov quotient is $T_{i+1}$ holds for $i=0$ and $i=1$.
A space is $T_2$ (Hausdorff) iff it is both $R_1$ and $T_0$. I have seen $R_0$ and $R_1$ used in some papers (in the context of uniform spaces, e.g.), but preregular not so much.