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i polished the definition in bundle gerbe and then reorganized the former material on “Interpretations” in a new section
that first shows how to get a shifted central extension of groupoids form the bundle gerbe, and then demonstrates that this is the total space of a principal 2-bundle
added to bundle gerbe more details on the proof of the fact that the bundle gerbe is the pullback
$\array{ P_{(Y,L,\mu)} &\to& \mathbf{E}\mathbf{B}U(1) \\ \downarrow && \downarrow \\ C(U) &\stackrel{\mu}{\to}& \mathbf{B}^2 U(1) \\ \downarrow^{\mathrlap{\simeq}} \\ X }$added two examples to bundle gerbe:
group extensions (=equivariant bundle gerbes over the point)
the construction of the “tautological bundle gerbe” from an integral closed 3-form on a simply connected manifold
I have added a pointer to the 1997 article by Gajer to bundle gerbe, bundle 2-gerbe and circle n-bundle with connection, where bundle $n$-gerbes for all $n$ pretty explicitly as $\mathbf{B}^{n+1}U(1)$-bundles (also with connection) is given.
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