The word “topological stack” has at least three usages:

 1. A stack $\mathcal{D}\rightarrow \text{Top}$ is said to be a topological stack  if there is a a morphism of stacks $p: \underline{M}\rightarrow \mathcal{D}$ for some manifold $M$, such that $p$ is a representable epimorphism. This is Definition 2.22, page number 86 in David Carchedi’s [thesis][1].
 2. A stack $\mathcal{D}\rightarrow \text{Top}$ is said to be a topological stack if there is a morphism of stacks $\underline{M}\rightarrow \mathcal{D}$ for a manifold $M$, such that $p$ is representable and has local sections. This is Definition $2.3$, page number 7 in Jochen Heinloth’s [Notes on Differentiable stacks][2].
 3. A stack $\mathcal{D}\rightarrow \text{Top}$ is said to be a topological stack  if there is a a morphism of stacks $p: \underline{M}\rightarrow \mathcal{D}$ for some manifold $M$, such that $p$ is a representable epimorphism and that it is a “local fibration”. This is Definition $13.8$, peg number $42$ in Behrang Noohi’s [Foundations of topological stacks, I][3].

There maybe more. Feel free to add if you know more.

  


  [1]: http://math.gmu.edu/~dcarched/Thesis_David_Carchedi.pdf
  [2]: https://www.uni-due.de/~hm0002/stacks.pdf
  [3]: https://arxiv.org/pdf/math/0503247.pdf