The groupoid representation of Joyal and Tierney may be identified with the truncated simplicial diagram appearing in the statement of Theorem 2 of the Lurie notes mentioned in the question. That is, a groupoid object is precisely a diagram of the shape indicated in Theorem 2 satisfying some axioms, which are automatically satisfied by construction when one takes pseudopullbacks as indicated in the diagram. In Joyal and Tierney's theorem, the groupoid in localic toposes is precisely this groupoid: it has topos of objects $U$, topos of 1-cells $U \times_X U$, and so forth.
To elaborate a bit, the diagram displayed in Theorem 2, which looks like
$U \times_X U \times_X U ^\to_\to \to U \times_X U ^\to_\to U$
(implicitly there are also some maps going back in the reverse direction) is really just the first few stages of the Cech nerve of the map $U \to X$. The Cech nerve as a whole is indexed by the dual simplex category $\Delta^{op}$, and here we just have the part on the 3 smallest objects of $\Delta^{op}$, namely $[0], [1], [2]$. It may be more familiar that when you truncate all the way to just two objects $[0],[1]$, you get the maps $U \times_X U ^\to_\to U$ -- the kernel pair of the map $U \to X$, which is an equivalence relation. The higher analog of the fact that a kernel pair is always an equivalence relation is that the Cech nerve of a map is always a groupoid object -- $[0]$ corresponds to the objects of the groupoid, $[1]$ corresponds to the morphisms, $[2]$ corresponds to composable pairs of morphisms (not to 2-cells), and in general $[n]$ corresponds to composable $n$-tuples of morphisms.
You can more generally think of an internal category object (not just an internal groupoid object) in a category as a certain type of simplicial object, as described here.