“But the Hawaii State Department of Health said Monday that there is no difference between a certificate and a certification of live birth in the eyes of the state.”
Notice the statement is missing BIRTH just before certificate. And in the eyes of the state, means next to nothing, but it does indicate there are two different documents.
There's no dispute about there being two documents. The "eyes of the state" is exactly what matters. It is the state that issues the documents, and the state that says what they mean.
The fact is, and what this quote says is, the COLB that was scanned is a legal document and is proof of the facts contained on it. A "vault copy" is not required.