Barry’s Certification of Live Birth is a short form - Abstract.
Whether wallet-sized, billboard-sized, or postage stamp-sized, it’s still a short form abstract and according to the Department of State website “may not be acceptable for passport purposes”.
No, it's not an abstract, in the sense used by the DOS. It fulfills all the qualifications listed by the DOS.
Futhermore, I think you'll find that more and more states, if not all, are now or will be issuing this type of document in lieu of a photostat of an original certificate in response to a birth certificate request. It saves time in not having to research microfilm or image files, and it saves money for the same reason.
As I said above, I have one, and it's been acceptable for a passport app, and I have no reason to believe it won't be acceptable for anything else that requires proof of birth in the US.