Use the APIAbort exception to defer the abort to _dispatch for consistency

[portante]

We should really use the APIAbort exception to defer the abort to _dispatch so we're consistent. We still have the user API classes that don't extend ApiBase, unfortunately, but they'll be gone soon. (I fixed PUT, and I'll retain that part of the PR along with the error-case functional tests when I can rebase on top of your NGINX fix.)

So I'll approve this without a lot of excitement; but it is an immediate fix for the namespace mismatch, and we still have the user APIs... and the message example is probably a nice touch.

Originally posted by @dbutenhof in #3130 (review)