My guess is that $X$ is a Riemann surface-with-boundary if it is a topological 2-manifold-with-boundary such that the transition charts are biholomorphic. Now what does biholomorphism mean for charts between half balls?
I presume a map $$f:\{\text{Im}(z)\geq 0\} \rightarrow \{\text{Im}(z)\geq 0\}$$ is said to be biholomorphic if it is a homeomorpism and it is biholomorphic on the interior.
Now, Schwarz reflection says $f$ can be extended to whole of $\mathbb C$.
Is this correct? If yes, is there a reference where this definition is made and used?