If HI included the sex (doubtful), then they would take a black marker and scribble the M out and hand write an F (or F for M). An adoption might or might not be different (every office does things slightly different) but there would be a big scribble over the original birth name. The clerk might try to squeeze in the adoptive name - under the proper alphabetical tab and chronologically between the other babies born before and after him. Or the clerk might enter the adoptive name at the end of that particular alphabetical section. White Out might also have been used, but that can sometimes be scratched off so big black scribbles worked best. So, you see, any changes or amendments would be obvious on the original handwritten index. But pages could be switched out easily if someone had a mind to...
I don’t know what the law required in 1961 for a handwritten birth index, but the law in Hawaii now requires the birth index to include the child’s name, gender, and date of birth.
As a rule, important information in a database is never deleted or changed. New or updated information is reflected by additional records.