Claudius is the personal version OF the genus "Claudii."
-General name, Family name, 'nickname', Personal name. <-The naming conventions of Patrician families in ancient Rome from circ. 30AD-on.
So, for an example: Justinian Claudius Catulus Titus.
Justinian is the name that others may use for him IF they have the proper rank. Claudius is his family name, because he is of the Claudii. Catulus is his nickname, that would be used among family. Titus is the very personal name that no one but family, close friends, or lovers would be allowed to use and that only with permission.