In [this note](https://ncatlab.org/nlab/files/Rezk_CompactlyGeneratedSpaces.pdf) a $k$-Hausdorff property is defined, between [weakly Hausdorff](https://topology.pi-base.org/properties/P000143) and [unique sequential limits](https://topology.pi-base.org/properties/P000099).

On the other hand, a different notion of $k$-Hausdorff between $T_2$ and [compacts are closed](https://topology.pi-base.org/properties/P000100) 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](https://math.stackexchange.com/questions/4760309/how-are-k-hausdorff-and-weakly-hausdorff-distinct).