Ha! Birth dates are a fun topic. People really didn't pay attention to those things especially in the old country. Even in the US they can be iffy.
A couple of examples: My gm was born in Warsaw, but she never really new her actual birthday. Somehow in the early 1960s one of her brothers got the date from Warsaw (I wish I knew how).
My father's New York City birth certificate has a "wrong" first name (as does my mother's) and a birth date that he said he didn't use as a child. The family celebrated it on a different day a few weeks earlier. Our theory is that the later date was the day the midwife registered the birth. He used the certificate date as his official b-day and celebrated it then too. And to make things crazier, his father's Petition for Naturalization, which lists his family, has a birth date for my father that is a year later and his sister's year was several years too early. And my adult father was there when the petition was filled out.
Unless there is a burning reason to have certainty for a birth date, I wouldn't get to worked up about the actual date. As long as you have the right person, be satisfied.