I know these two groups are the same (the group of orientation-preserving isometries of the hyperbolic plane), but I didn't manage to find an explicit isomorphism.
Could you explain me how to find one, and maybe a geometric interpretation of this action ?
In other words, how the elements of $SL_2(\mathbb{R})$ act on the upper hyperboloid surface ? (for example, what is the geometrical action of an hyperbolic element, an elliptic element and a parabolic element of $SL_2(\mathbb{R})$ on the hyperboloid ?)
The Lie group $SL_2(\mathbb{R})$ acts on its 3-dimensional Lie algebra $\mathfrak{sl}_2(\mathbb{R})$ via the adjoint action, preserving the Killing form which has signature (1,2). This gives an action of $SL_2(\mathbb{R})$ on $\mathbb{R}^{1,2}$, which you could (painstakingly) write out explicitly.