It is certainly not “false informaton”, because it certainly does not claim to be anything more than what it is, the assignment of parental responsibility, regardless of the biological circumstances.
Yes yes, all very lawyerly. Ordinary people don't couch their words in legal jargon, they want straight answers. A straight answer is that an Amended birth certificate contains FALSE information.
The Father listed on my official BIRTH certificate, is not my BIRTH father. You can say "assignment of parental responsibility", but most people assume it to mean "BIRTH" father. The term "Birth" on the Certificate implies an Adjective Modifier of every term contained therein.
The usual and ordinary Rules of English compel everyone to associate every term on that Legal document with the incident of "Birth", not some legal hearing which may or may not have occurred at some later date, and which the State (the originators of the document) won't even confirm or deny.
And that is the INTENDED purpose of these legal machinations. To conceal the truth.