Is there a modern generally accepted answer regarding the notions of k-space or compactly generated space?
For example there are currently at least 3 formally distinct notions of k-space in wide circulation:
In Kelley's General Topology, $X$ is a k-space if for $S \subseteq X$ not closed in $X$ there is a closed compact subspace $C \subseteq X$ such that $C \cap S$ is not closed in $X$.
(This notion of k-space also appears in A. Wilansky's _Between T1 nd T2 (Amer. Math. Monthly, vol.74, no.3, pp.261-266).)
According to nLab, $X$ is a k-space if whenever $S \subseteq X$ is not closed in $X$, there exists a compact Hausdorff space $K$ and a map $f:K \to X$ such that the preimage of $S$ is not compact.
This is equivalent to $X$ being compactly generated (CG) in Neil Strickland's note The category of CGWH spaces.
Wikipedia declares that $X$ is a k-space (or a compactly generated space) provided that whenever $S \subseteq X$ is not closed in $X$, then there exists a compact subspace $C$ of $X$, such that the intersection of $C$ and $S$ is not compact.
Are any of definitions 1,2,3 equivalent if $X$ is not weakly Hausdorff?