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).
In Kievan Rus, the common form of Vladimir was Volodimir. Ukraine still has this form (Volodymyr Zelensky).
The diminutive of Volodimir was Vova, which makes sense. Later, in Russia, Volodimir was replaced with the Church Slavonic form, Vladimir.
So today diminutive of Vladimir is Vova (not Vlad, a common mistake in the West to call Putin Vlad).