I’m not so sure that you can say any state issues a certificate of live birth.
A Certificate is signed by the attending physician under penalties of perjury that he delivered a live child on such and such a date.
It is witnessed and attested to by the hospital registrar, or another witness.
So it seems to me that it is more of an affidavit that the state recognizes - not a document that the state generates.
NO, that would be called a "Letter of Verification." Also, don't confuse "Certificate" with "Certification." Ther former is the original BC, the latter is a computer-generated form, but it is what Hawaii, and most states, issues as a birth certificate. Except in certain cases, the "Certification of Live Birth" is what Hawaii will send you when you request a copy of your birth certificate.
It can be used like a regular birth certificate, but it will not prove where one was born as they issued these to foreign-born children.