Hawaii will give birth certificates to the children of residents regardless where they are born so long as the parent requests a birth certificate within a year of the birth.
The doctor that examines the child during that year will be listed as the birth doctor.
I looked up the Hawaii statutes many years ago.
A Hawaii birth certificate is not proof that you were born in Hawaii.
I think that was the “Certificate of Hawaiian Birth,” a program that got dropped at some point, not the “Certificate of Live Birth” certified by a hospital and doctor. The CHB was for people who had been born at home, though others took advantage of it as well (Chinese leader Sun Yat-Sen, for example).