Are any of these notions of "k-space" equivalent if $X$ is not assumed weakly Hausdorff? 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?
 A: I think the Wikipedia definition is not the best, since it does not deal nicely with the non-Hausdorff case, and a quotient of a Hausdorff space need not be Hausdorff. This is kind of related to the question of whether locally compact means each point has a compact neighbourhood, or has a base of compact neighbourhoods. The latter concept is more in tune with the notion of a local property.
A general discussion of "Monoidal closed categories of spaces" is in a paper by Booth and Tillotson (Pacific J Math, vol 88)   available here.
Section 5.9 of my book Topology and groupoids (as in the 1988 differently titled edition) has the following result:
5.9.1 Let $X$ be a space.  Then the following are equivalent:
(a) $X$ is a $k$-space;
(b) there is a set $\mathcal C_{X}$ of maps $t : C_{t} \to X$ for compact Hausdorff
spaces $C_t$ such that a set $A$ is closed in $X$ if and only if $t^{-1}(A)$ is
closed in $C_{t}$ for all $t \in \mathcal C_{X}$;
(c)
$X$ is an identification space of a space which is a sum of compact Hausdorff
spaces.
So the n-lab definition agrees with this. Note that this Section also considers the convenient category of $k$-continuous functions,  using the test-open topology on spaces of k-continuous maps.
Actually the idea of fibred exponential laws (i.e. some notion of locally cartesian closed) comes from a paper of Thom on "Homologie des espaces functionels" but the details were sketchy, and were developed by Peter Booth.
A: The three definitions are not equivalent for general topological spaces.
As shown in this question, Definition 2 (nlab) implies Definition 3 (Willard), but not conversely.  For example the one-point compactification of a space which does not satisfy Definition 2 (like the Arens-Fort space) is compact and hence satisfies Def 3, but does not satisfy Def 2.
Both classes of spaces with Def 2 and with Def 3 are stable under taking quotients, and neither requires the Hausdorff condition.  But Def 2 is slightly nicer in that it is stable under taking open sets (see here).
As for Def 1 (Kelley), it seems that it's the least "natural" of the three.  Def 1 (resp. Def 3) can be defined as the topology being coherent with the collection of closed compact subspaces (resp. compact subspaces).  Being a compact subspace in a space $X$ is an intrinsic property of the subspace. But being compact and closed is not, it depends on how the subspace embeds into $X$.  And furthermore, any space is covered by its compact subspaces (which include all singletons).  But there need not even exist a covering of the whole space by closed compact subspaces.  For example, in the particular point topology with an infinite number of points and particular point $p$, there is no closed compact set containing $p$.
Note that Wilansky's article used Def 1 of k-space, but only in the context of KC spaces (KC = all compacts are closed).  For those spaces, Def 1 and Def 3 are equivalent.
