Birth certicates should be reserved for biological information for the obvious biological/health/scientific reasons. It is stupid to deliberately create biologically inaccurate or incomplete birth records.
It is also disparate treatment. Non-biologically related (to the child) hetero parents have to adopt to become legal parents, why should't the non-bio related gay parent(s) have to adopt also.
According to Indiana law, My name was listed on the revised birth certificate. There is nothing on the document to indicate my sone is adopted.