I don't have a reference, but the following parametrization may help to describe this moduli space. For each polygon in your sense, there is a representation by the Schwarz-Christoffel formula $$f(z)=C\int_a^z\prod_{k=1}^n (\zeta-a_k)^{\alpha_k-1}d\zeta,$$ where $f$ is a Riemann map of the upper half-plane onto your polygon, $C\neq 0$ and $a$ are arbitrary constants; $\pi\alpha_j>0$ are inner angles subject to $$\alpha_1+\ldots+\alpha_n=n-2,$$ and $a_j$ are distinct points on the real line. The polygons are equal if the corresponding sequences $(a_1,\ldots,a_n)$ correspond by an element of $SL(2,R)$.
The description is simplified if you set one prevertex to be $\infty$. Then you have $n-1$ tuples $(a_1,\ldots,a_{n-1})$ modulo the action of the affine group $x\mapsto cx+d$.
Hyperbolic polygons have a similar description, but without a formula: to each sequence of angles and an equivalence class of pre-vertices (images of vertices under a Riemann map) there exists exactly one polygon if the angles are subject to the restriction $$\sum_{j=1}^n\alpha_j< n-2.$$ This is a very old result going back to E. Picard.
Remark. To derive a parametrization of polygonal regions from Kapovich Milson seems hopeless, since the condition of existence of continuation of a "piecewise-immersion" of the circle to the disk is too complicated to handle. Gabrielov and I tried this for the much more complicated case of spherical polygons, and this does not work. On this subject, you may look at https://www.math.purdue.edu/~gabriea/s-klein.pdf