Topological embeddings are regular monos in the category of topological spaces. Topological immersions are arrows which are locally embeddings, i.e it's possible to restrict to an open cover such that each restriction is a topological embedding.
What about topological submersions? What's their categorical definition?
The symmetric definition would be maps that are locally regular epimorphisms. The regular epimorphisms of the category of topological space are quotient maps. This answer notes surjective smooth submersions possess the smooth analogue of the universal property of quotient maps, but are not characterized by it (not all arrows possessing this universal property are surjective smooth submersions).
The nlab page lists the general definition as an arrow such that every point of the domain is covered by a local section. This is discussed in this MO question but I can't make much out of it. Since the tangent-space definitions of smooth immersions and submersions are dual, I was hoping for something along those lines in the topological case.
What's the correct definition?
First, I don't think one can call something the "correct definition". Different definitions will be suitable for different authors in different circumstances.
One reasonable definition of a topological submersion (which I pulled from F. Lin, "A Morse-Bott approach to monopole Floer homology and the Triangulation conjecture") is as follows.
(This is quite similar to the suggestion in Tom Goodwillie's answer in the question you linked.) Then one would say that $\pi$ is a topological submersion if it is a topological submersion with respect to any choice of $q_0$ and with $S_0 = \pi^{-1}(q_0)$. This is more or less the statement that "locally in the domain and codomain, $\pi$ is a fiber bundle projection." This, to me, is the topological essence in what a submersion is. To further back this definition up, a CAT submersion (where CAT = TOP, PL, DIFF) is defined in Kirby-Siebenmann to be a map $f: X \to Y$ such that near every point $x \in X$, there is a neighborhood $U$ with $f(U)$ open and $U \to f(U)$ isomorphic as a map in CAT to the projection map of a product (aka, the map is locally a trivial fiber bundle). This is precisely Lin's definition for all $q_0$ and $S_0 = \pi^{-1}(q_0)$. If you're interested in the notion of transversality, Kirby-Siebenmann have a definition of that, though it's somewhat complicated and uses the notion of microbundles.
The definition you give, that through every point $x \in X$ there is a local section of the map with $x$ in its image, is not equivalent (though of course Lin and Kirby-Siebenmann's definitions imply it). The map $f: \Bbb R^2 \to \Bbb R$, $f(x,y) = xy$, satisfies your definition; a section passing through $(x_0, y_0)$ with $y_0 \neq 0$ is given by $s(t) = (t/y_0,y_0)$, and similarly for $x_0 \neq 0$; for $(x_0,y_0) = (0,0)$ take $s(t) = \text{sgn}(t)(\sqrt{|t|},\sqrt{|t|})$. But $f$ is not locally a fiber bundle projection near $(0,0)$.
Personally, I don't think that example deserves to be a topological submersion. But to even out my discussion, I'll point out that even in my definition of topological submersion, the preimage of a point under a topological submersion of manifolds doesn't need to be a manifold. To prove this, you just need an example of a topological space that's not a manifold $X$ and a manifold $M$ such that $M \times X$ is a topological manifold. These are reasonably abundant; there are examples with $M = \Bbb R$ and $M \times X = \Bbb R^4$.