It is well known that geometrically finite Fuchsian groups, or finitely generated discrete subgroups of $\textrm{PSL}_2(\mathbb{R})$ can be classified up to isomorphism by their signature $[g,s;m_1,\dots,m_r]$ which define a presentation
$\Gamma = \left\langle a_1,b_1,\dots,a_g,b_g,c_1\dots,c_s,d_1,\dots,d_r\ |\ d_1^{m_1}=\dots=d_r^{m_r}=\prod_{i=1}^g[a_i,b_i]\prod_{j=1}^{r}d_j\prod_{k=1}^sc_k=1 \right\rangle.$
Does there exist a similar presentation for discrete subgroups of $\textrm{PGL}_2(\mathbb{R})$?
Geometrically speaking, $PGL(2,{\mathbb R})$ is the full isometry group of the hyperbolic plane ${\mathbb H}^2$. NEC (Noneuclidean crystallographic) groups are discrete isometry groups of ${\mathbb H}^2$ acting cocompactly.
You may want to read Macbeath's paper "The Classification of Non-Euclidean Plane Crystallographic Groups"; it contains presentations of NEC groups and much more. The issue, however, is that the paper deals only NEC groups. One can write down a general presentation for geometrically finite isometry groups of the hyperbolic plane by taking Macbeath's presentations and declaring some of the exponents $m_i$ to be zero. (There will be a few exceptional cases that will be missing: These are finite extensions of cyclic groups.) The proof that this is a complete list of presentations will be essentially the same as in Macbeath's paper.