Equivalent definition of fundamental region

Question

I'm reading the paper "The fundamental region for a Fuchsian group" by L.R. Ford, where he seems to use the following definition of a fundamental domain (section 6 of the paper):

Let $\mathcal{U}$ denote the Poincaré disk model of the hyperbolic space.

Def.: A fundamental domain for a Fuchsian group $\Gamma$ is a closed region $\mathcal{F} \subset \mathcal{U}$ such that

1. no two interior points of $\mathcal{F}$ are congruent;
2. any region adjacent to $\mathcal{F}$ in $\mathcal{U}$ contains some points congruent to points in $\mathcal{F}$.

I tried to show that this is implies that $\bigcup_{\gamma \in \Gamma} \gamma.\mathcal{F} = \mathcal{U}$, but didn't succeed. Does Ford use a different definition of a fundamental domain than the one common today? Or are they equivalent?

Thanks!