This is very interesting, I have often wondered about this.
DERIVATIVE: I haven't read this word in an name book and I wouldn't use it myself. It seems indeed too general and if I used it then only to describe the connection between a name and a vocabulary word.
DIMINUTIVE: I think, it's not a shorter form of a name, but rather the name for a shorter bearer, of you know what I mean. In fact diminutives are usually longer, because specific syllables such as -ke or -ette are added to the name.
PET FORM: I would say a pet form would have to have something to do with the real name (even though some changes are hard to follow, like Mary to Polly). It's still the same name, but it was changed (not neccessarily shortened) to show affection towards the bearer (usually within a family or among good friends).
NICKNAME: It doesn't have to be related to the real name, but it can. It can show affection, but also derision. (Going by the etymology of the word nickname, it should be an additional name.)
VARIANT: This always sounds rather vague to me, any variation could be called a variant.