The definition of a gerbe on a smooth manifold that I know is that - after fixing an open cover $U_i$, a gerbe consists of the data of line bundles $L_{ij}$ on two-fold-intersections $U_{ij}$, isomorphisms $\alpha_{ijk}: L_{ij} \otimes L{jk} \longrightarrow L_{ik}$$\alpha_{ijk}: L_{ij} \otimes L_{jk} \longrightarrow L_{ik}$ on three-fold intersections that satisfy a co-cycle condition on four-fold intersections.
A gerbe on a site is a stack $G$, such that for every object $U$, there exists a covering $U_i$ of $U$ such that $F_{U_i}$ is non-empty for every $i$ and for any two objects $x_1$, $x_2$ in $G_{U}$, there exists a covering $U_i$ of $U$ such that $x_1|_{U_i}$ and $x_2|_{U_i}$ are isomorphic (i.e. objects exist locally and they are locally isomorphic).
My question is that if these two notions are related or if it is just the same name for completely different things. In particular: Are gerbes on a manifold a special stack on the small site of that manifold? Is there a fully faithful functor of $2$-categories that sends Gerbesgerbes over $M$ to stacks over (the small site of) $M$?
