I am more familiar with Cartan geometry, and in this setting we have a notion of development of curves. As described in Cap & Slovak 1.5.17, on a Cartan geometry $(\mathcal{P} \to M, \omega)$ modelled on $G/P$, we have a "Cartan space" which is an associated bundle $S :=\mathcal{P} \times^P G/P \to M$ that comes equipped with a canonical section $O: x \mapsto [u, eP]$ as well as a connection induced by the Cartan connection. Then there is a development map which takes germs of curves in the base $M$ to germs of curves in the model $G/P$ (technically the fibre of the Cartan space until we pick a frame in the $\mathcal{P}$). Explicitly, for a (germ of a) curve $c$ in $M$ with $c(0)=x$, we get (a germ of a) curve $\mathrm{dev}_c$ in the fibre $S_x$ of the Cartan space given by $$\mathrm{dev}_c(t) := \mathrm{Pt}_{c_t}((O(c(t)),-t)$$ where $c_t(s) := c(t+s)$. After picking a frame $u \in S_x$ we can identify $S_x$ with the model $G/P$.

Since n-dimensional (oriented) Hyperbolic geometry can be viewed as a Cartan geometry modelled on $\mathrm{SO}(n,1)/\mathrm{SO}(n)$, we also have this notion of development of curves in hyperbolic geometry.

But in the hyperbolic geometry literature (which I am mostly unfamiliar with) there is another notion of developing/development: For $(M,g)$ a connected hyperbolic $n$-manifold, there is a developing/development map on the universal cover $\mathrm{dev}: \widetilde{M} \to \mathbb{H}^n$ which is an isometry if $M$ is complete (if I remember correctly).

Is there some nice way to related these two notions of development? Or is the naming mostly coincidence?