The new vocative has nothing to do with the old vocative (whose forms would've been *Маше, *Зино and *Димо, indistinguishable by ear from the nominative but probably reflected in writing).
If we are to count all such case-like forms limited to a single paradigm and/or context, we'd end up with quite a few: partitive (чашка чаю vs. вкус чая), locative (...
I totally agree with the answer Nikolay provided, I just want to add one other important points made by opponents of calling this new forms vocative case, here's a quote:
Основное различие – с существительными в новой звательной форме, в
отличие от звательного падежа в древнерусском и других славянских
языках («Боже правый»), невозможно согласовать ...
Кать! and Саш! (sic, no soft mark) are the so-called new vocative, which is has no relation to the original Slavic one. And, outside of names, you also have it in мам!, пап!, дядь!, and тёть!
Господи is an old vocative. Старче (from старец) is an old vocative that's barely ever used, but widely known thanks to Pushkin's Tale of the Fisherman and the Fish.
As for old vocatives, there are also Иисусе (more often господи Иисусе) and врачу, исцелися сам are used. As for the the last idiom, many Russians don't understand that врачу is an old vocative, instead interpreting it as a dative case.
As for the new vocatives, they are created from virtually any name which ends by -a or -я (Саш, Дим, Зин, Вась, Петь, ...
I believe that one of answers already given should be accepted, since all of them are succinct, complete and correct. Still, I want to mention a couple of other examples, so, there's a word владыко which is vocative of "владыка". It's used to address higher ranks church officials, so domain of usage is pretty tiny. In fact, in Russian is extinct to that ...
Not exactly names
From slang there's the fringe чУви (from чувак - dude)
Then братИш, братЮнь (from братиш(к)а and братюня respectively - bro/bruh)
ПарнИш (from парниша - bloke/lad)
ДевУль (from девуля - girl/gal)
СынУль (from сынуля - sonny)
all formed by the pattern of the new vocative
Exclusively for сын and дочь there're in use such vocative ...