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.
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.
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.

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