The name on the certificate is Kapiolani Maternity and Gynecological Hospital. The hospital took that name in 1931. In 1978 it merged with the Kauikeōlani Children’s Hospital to become the Kapiʻolani Medical Center for Women & Children, which it is now. There was nothing wrong with the name of the hospital in the birth certificate.
It doesn’t look like there was any bar against resident foreigners attending Indonesian schools, and in Obama’s case, one of the schools he attended was private and Catholic. A Parent can’t renounce her child’s US citizenship. It would have been very foolish for an American to renounce US citizenship in 1960s Indonesia. It was a very violent time.
I have a “certificate of live birth”. I was adopted. My original birth certificate is locked away somewhere. My biological parents’ names are on it.
On my Certificate of Live Birth, my adopted parents’ names are on it.
That in itself signifies that it’s not his original birth certificate with his real, biological parents named.