The literal translation for this sentence is "Me they call Vanya" "Меня" being the accusative form here.
Do Russians actually use the diminutive forms of their names when introducing themselves?
Yes, in a proper situation (when it's not too formal and official, they are relatively young, it's a friendly meeting etc)
Kids on a playground was what this sentence suggested to me, I guess that would be an appropriate interpretation?
Kids on the playground for sure, what I meant to say is that it is not only for children.
So, i would use this to introduce myself to somebody in a friendly manor or is ot still considered childish?