You are correct, I believe. This isn't a gay issue.
If you want to keep it secret, have a "for public use" record which has what the parents want. But lets make sure the OFFICIAL DOCUMENTS tell the truth.
I completely agree.
Both the new birth certificate and the original birth certificate are official documents- in most states, the original is sealed after adoption, and depending on state law a child may or may not have a right to see it as an adult. In many states, you cannot ever see your original birth certicates without a court order and/or consent from your birth parents.