Birthday vs Birthdate

It makes more sense to me that the day we celebrate our birth be called our Birth Date, like "Happy Birth date" rather than "Happy Birthday", Birthday just makes me think of the exact DAY you were born, you will never be born again, that should be your BirthDAY. Then, for the rest of our lives we celebrate our Birth DATE, the DATE on which we were born. On legal information they ask for Birth date, implying they want the year, month, the whole shebang. Again, BirthDAY seems to work better here for me because it is the only DAY you were born.