so the whole Claudian clan (Tiberius, Caligula, Claudius, and Nero) got saddled with a nom de guerre, as it were?

Nope. Tiberius was a member of the Claudius gens. Tiberius was the son of Tiberius Claudius Nero. Tiberius was Augustus' stepson, because his mother Livia Drusilla married Augustus. That's how the Claudians got merged with the Julian gens (of Caesar fame) to name the Julio-Claudian emperors.

Ceci n'est pas un seing.