I would have said that "surname" as a replacement for "last name" would be the better choice, but on reflection, "family name" is probably the best -- and most proper for genealogy, where we are, after all, concerned with families.
Surnames were often acquired AFTER a person was born, i.e. John the Smith, John the Butcher, so they might figure under "current name" but not "name at birth." Except when the surname then became the family name for subsequent generations.
Family name also can include names which were not really "last names at birth" but an appropriate way to identify someone, such as the current LNABs we've assigned to many very early profiles.