I have been learning about Fuchsian groups, and from what I understand (based on the notes here: https://math.la.asu.edu/~paupert/CriderPoincarePolygonTheorem.pdf), there is a group of symmetries of the hyperbolic plane generating a regular octagonal tiling with presentation
$$ \langle g_1,g_2,g_3,g_4 \mid g_4^{-1}g_3^{-1}g_4g_3g_2^{-1}g_1^{-1}g_2g_1=1\rangle. $$
I decided to search numerically for subgroups of $SL(2,\mathbb{R})$ with this presentation, and found elements $g_1,g_2,g_3,g_4$ satisfying the relation above but which did not appear to be Fuchsian (I plotted the orbit of a point and saw that as I went to longer words, the orbit seemed to get dense).
If my numerics and my understanding is correct, this means it might be possible to have two groups $G,G'\subset SL(2,\mathbb{R})$ such that $G$ is Fuchsian and $G'$ is isomorphic to $G$ (as a group), but $G'$ is not Fuchsian. Is this correct, or have I made a mistake?
In other words, to state the question cleanly: Is it possible to find two groups $G,G'\subset SL(2,\mathbb{R})$ with the same presentation, such that $G$ is Fuchsian and $G'$ is not?