No, in Hawaiian usage, the Certificate is the original "long form" version of the Birth Certificate. The Certification of Live Birth is an electronic abstract or "short form", containing only part of the information on the original "long form> although in some places even the abstracts were once hand generated from birth "Logs", rather than being copies of the original BC. The first BC of mine that I got was clearly printed from a Microfilm or microfiche copy of the original. So was a copy of our marriage certificate obtained in the mid 1980s.
Before all the controversy over BHO's BC erupted, I had read that many states were going "paperless", and would store computer images of the original documents, and print those, eliminating the need for the absract. Doing that requires a lot more storage space, but that is getting very very inexpensive. The abstract forms were "invented" to save time, and money. The minimal data they contain, different from state to state, was stored as "characters" not images, and thus took up very little storage space. By not having to retrieve the physical document, much money could be saved. Proably to spend on AFDC or similar programs.
I’ve ordered a couple of birth certificates that were the long form, and apparently had been printed from microfiche. They were not abstracts with fields pulled from a database.