Someone posted the birth certificate of a person born in 1930
here. It does give the "race" of the parents, but it looks as though the registrar took the parents at their word. If they said "White" he or she wouldn't write down "Caucasian" or vice versa or even make the races of a White and a Caucasian parent the same. So if the parents said "African" they most likely wouldn't change it to "Negro" or "Black" or "colored."
I don't know when the birth certificates stopped listing race, but a lot happened in the sixties so far as race and color were concerned. At some point in the 1960s states may have stopped requiring this information -- or they may have continued to keep track of the data for statistical reasons but kept it off of birth certificates and other legal papers issued to the public.
There was a period of time (64-67 or thereabouts) where it quickly became the practice to NOT include "race". Then federal law began requiring reports concerning open accommodations, home loans, etc. showing "race" so that it could be determined if "race" was used to discriminate against people.
So, yes, not just the nomenclature used for "race", but also the very question itself changed, and in dramatic ways.
The colorblind society was safely tucked out of the way so that the racebaiting welfare pimps could make a living.