In Rezk - Compactly generated spaces 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. Lawson and Madison - Quotients of k-semigroups.

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.