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In the state of Hawaii, a certificate of live birth and a birth certificate are the same thing.

No. They are not. Hawaii has 4 different types of certificates. They are not the same.

You wrote... “No. They are not. Hawaii has 4 different types of certificates. They are not the same.”
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Every long form Hawaiian birth certificate I’ve seen says Certificate Of Live Birth on top.

Thank you, and quite right.
A COLB does not list the delivery doctor
It does not list the hospital, or the administrator, it does not have footprints, or anything more than a short form version of the Long Form, which is why you need a long form to remove doubt concerning running for POTUS.

“The Territorial Public Health Statistics Act in the 1955 Revised Laws of Hawaii
In the State of Hawaii, back in 1961, there were four different ways to get an “original birth certificate” on record.
BC1. If the birth was attended by a physician or mid wife, ....
BC2. In 1961, if a person was born in Hawaii but not attended by a physician or midwife, then all that was required was that one of the parents send in a birth certificate to be filed. The birth certificate could be filed by mail.
.... there is and was no requirement for a physician or midwife to witness, state or report that the baby was born in Hawaii.
BC3. In 1961, if a person was born in Hawaii but not attended by a physician or midwife, then, up to the first birthday of the child, a “Delayed Certificate” could be filed...
BC4. If a child is born in Hawaii, for whom no physician or mid wife filed a certificate of live birth, and for whom no Delayed Certificate was filed before the first birthday, then a Certificate of Hawaiian Birth could be issued upon testimony of an adult (including the subject person [i.e. the birth child as an adult]) if the Office of the Lieutenant Governor was satisfied that a person was born in Hawaii, provided that the person had attained the age of one year. ...

In 1982, the vital records law was amended to create a fifth kind of “original birth certificate”. Under Act 182 H.B. NO. 3016-82, “Upon application of an adult or the legal parents of a minor child, the director of health shall issue a birth certificate for such adult or minor, provided that the proof has been submitted to the director of health that the legal parents of such individual while living without the Territory or State of Hawaii had declared the Territory or State of Hawaii as their legal residence for at least one year immediately preceding the birth or adoption of such child.”
In this way “state policies and procedures” accommodate even “children born out of State” (this is the actual language of Act 182) with an “original birth certificate on record.”