The usual classification of covering spaces (stated in terms of a categorical equivalence) requires the conditions "locally path connected" and "semi-locally simply connected."
**Theorem 1:** If $X$ is a path-connected, locally path-connected and semi-locally simply connected space, then the category of covering spaces over $X$ is equivalent to the category of left $\pi_1(X)$ actions on sets.
If $X$ is locally path connected, the fiber bundles with totally path disconnected fibers are actually in bijective correspondence with the (genuine) coverings of $X$, that is, any such fiber bundle corresponds to a unique covering map. To see this we can exploit the fact that these fiber bundles cannot possibly be locally path connected unless they are already covering maps.
Given any space $Y$, let $lpc(Y)$ be the underlying set of $Y$ with the topology generated by the path components of open sets of $Y$. This topology is finer than the topology on $Y$ and the continuous identity function $c_Y:lpc(Y)\to Y$ is universal in the sense that if $Z$ is locally path connected and $f:Z\to X$ is continuous, then so is $f:Z\to lpc(X)$. In other words, $lpc$ is left adjoint to the inclusion of locally path connected spaces in to all topological spaces and $c_Y$ is the counit.
Since $Z=[0,1]^n$ is locally path connected, $lpc(X)$ is path connected whenever $X$ is (since both topologies admit exactly the same paths) and $lpc(X)\to X$ induces an isomorphism $\pi_1(lpc(X),x)\to \pi_1(X,x)$ for any $x$.
Now if $p:E\to X$ is a fiber bundle with totally disconnected fiber $F$, then there is a unique covering map $p':lpc(E)\to X$ such that $p'=p\circ c_{E}$. If we forget the topology on $F$, then the $\pi_1(X)$-action from the bundle gives the $\pi_1(X)$-action that characterizes the covering $p'$.
So if $X$ is path connected and locally path connected, the category of coverings over $X$ and the category of fiber bundles over $X$ with totally path-disconnected fiber are equivalent. They must "embed" into the category of $\pi_1(X)$-sets, however, neither must be equivalent to the category of $\pi_1(X)$-sets unless $X$ is semilocally-simply connected.
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I do will do my best to address your question about the other generalizations of covering space theory, however, there are a lot of them so I will probably miss a few. I would suggest looking at some of the reference in the papers I'll mention below. I should clarify with this: **There is no "correct" generalization of covering space theory**. The usefulness of a given generalization depends on the intended application and these vary greatly.
1) **Covering maps:** There are ways to classify covering maps (in the usual sense) of arbitrary locally path connected spaces. One is to use shape theoretic/steenrod homotopy methods for finite sheeted coverings or one can use topologized versions of the fundamental group to classify arbitrary coverings. For instance in [2] and [3] it is shown that open subgroups of $\pi_(X)$ (with a certain topology) which contain and open normal subgroup classify coverings of $X$ even when $X$ is not semilocally simply connected.
For finite sheeted coverings see:
[1] L. Javier Hernández Paricio and Vlasta Matijevic, Fundamental groups and finite
sheeted coverings, J. Pure Appl. Algebra 214 (2010), no. 3, 281–296.
For arbitrary coverings see:
[2] Hanspeter Fischer, Andreas Zastrow, A core-free semicovering of the Hawaiian Earring, Topology Appl. 160
(2013), no. 14, 1957–1967.
[3] J. Brazas, Semicoverings, coverings, overlays and open subgroups of the quasitopological fundamental group, Topology Proc. 44 (2014) 285-313.
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2) **Overlays:** One of the most recognized generalizations of covering space theory is Fox's theory of overlays. Overlays are "nice" covering maps where you can uniquely lift chains of neighborhoods in a trivializing cover (this is not built into the definition of covering map). Overlays do not provide more insight into $pi_1$ but the reason they are so important is that they admit a beautiful classification for all compact metric spaces in terms of actions of the symmetric group on the fundamental pro-group.
[4] Ralph H. Fox, On shape, Fund. Math. 74 (1972), no. 1, 47–71.
[5] Sibe Mardešic and Vlasta Matijevic, Classifying overlay structures of topological
spaces, Topology Appl. 113 (2001), no. 1-3, 167–209.
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3) **Semicoverings:** Ok, so I am a little biased on this one, but semicovering admit a categorical classification just like the one for coverings. Semicoverings are much more general than coverings and the classification applies to all locally path connected - and even some non-locally path connected - spaces. Here $\pi_1(X)$ has a certain topology and the category of semicoverings over $X$ is equivalent to the category of continuous $\pi_1(X)$-actions on discrete spaces. In otherwords, open stabilizer subgroups classify the semicoverings. The upshot is that this has led to solutions to long-standing open subgroup problems in the theory of topological groups [7].
[6] J. Brazas, Semicoverings: A generalization of covering space theory, Homology Homotopy Appl. 14 (2012), no. 1, 33–63.
[7] J. Brazas, Open subgroups of free topological groups, To appear in Fund. Math. 2014.
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4) **Generalized coverings defined only in terms of unique lifting properties:** Introduced by Hanspeter Fischer and Andreas Zastrow for locally path connected spaces in their amazingly insightful paper [8], these have been used more recently to gain insight into the algebraic structure of fundamental groups of wild spaces. These behave just like covering maps in nearly every respect except local triviality (or even being a local homeomorphism). There is no "nice" categorical classification of these due to the fact that the general occurence of unique path lifting property is difficult to characterize.
[8] Hanspeter Fischer and Andreas Zastrow, Generalized universal covering spaces
and the shape group, Fund. Math. 197 (2007), 167–196.
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There are so many other approaches (for instance a number of theories for uniform spaces) and there is no way I will hit them all here so I suggest looking at the references in these papers to see what has been done. I know I try to give a general overview in the introduction of [3].