Oh that's interesting. Explains the other common diminutive of Vladimir - Volodya. And yeah, Vlad is short for Vladislav.
My favorite set of Russian diminutives that makes zero logical sense to a non-native is: Alexander (formal) = Sasha (informal) = Shura/Shurik (very informal).
My favorite set of Russian diminutives that makes zero logical sense to a non-native is: Alexander (formal) = Sasha (informal) = Shura/Shurik (very informal).