It seems to me that the guidelines they reference are only about how the vital statistics are reported to the Feds. I don’t see anything that says the states have to record the place of birth that way on the certificates they issue. What says they didn’t record the actual place of birth on the certificate, and just use the mother’s residence when they transmitted the (presumably summarized) statistics to the Feds?
I know, for example, that my birth certificate lists my actual place of birth, which was not ever my mother’s place of residence.
That's my take as well. The birth certificate would still show the correct place of birth.