Kazimierz [ka-zhi-miezh] formal
Kazik [ka-zhik] pet-name
Kaziu [ka-zh-yu] - vocative case of pet-name
Kazimierz is formal, right.
Kazik is a pet-name
Kaziu is _not_ the vocative of Kazik (which would be Kaziku) it's the vocative of Kazio, another pet-name.
In standard informal spoken Polish the vocative can be used a nominative but this only works for pet-names for men/boys where the vocative ends in -u, when the vocative would end in -(i)e you can't do that and if it's not a pet name you can't.
Leszku could be the vocative/nominative for Leszek as a petname for Lech but not for the separate man's name Leszek.