Actually it is the other way around. He could have gotten a Certificate of Live Birth (COLB) from HI after the fact, and that is the (allegedly forged) document that was briefly posted on his site. He would only get a Birth Certificate if he was born in the state.
Let's keep our terminology straight. The Certificate of Live Birth is the original "Birth Certificate", usually signed by the doctor or nurse/midwife and a hospital official. In the case of "after the fact" registration, it would show the names of witnesses or others with direct knowlege of the circumstance of the birth. It should in either case show the proper city, state and/or country of birth. It's also known as the "long form" birth certificate, and the original would be the "Vault copy". It can however be modified, as in the case of an adoption or legal name change.
The Certification of Live Birth, OTOH, is a "generated on request" extract of data from the state's birth records database. It is AKA the "short form" birth certificate. If legitimate, it would also show place of birth. In the old days, if generated at all rather than just making a copy of the original "Certificate", it would be produced by manually extracting data from the original "Certificate". Now however it is a computer generated form, and thus much easier to fake, with the information extracted from the electronic records in the state's database.