In this note a $k$-Hausdorff property is defined, between weakly Hausdorff and unique sequential limits.

On the other hand, a different notion of $k$-Hausdorff between $T_2$ and compacts are closed was already in use in the literature, e.g. https://doi.org/10.1007%2FBF02194829.

Has the weaker $k$-Hausdorff property appeared in the literature beyond that nlab note? The preamble seemed to indicate it was novel. See also some discussion at Math.SE.