As far as I know it, they go like this. Unfortunately, these words tend to cross over quite a bit--some are basically synonyms. It's hard to put an exact definition for these, but that would be a very good idea for your book. Right in front, explain the way you're using each term.
Derivitive: The definition of a derivation is this: "the formation of a word from another word or base (as by the addition of a usually noninflectional affix)" (m-w.com). So a name derivation would be a name that comes from another. I think that this could be considered a catch-all for the other terms, the way that alliteration covers both assonance (repeated vowel sounds) and consonance (repeated consonant sounds).
Diminutive: This would be the shorter version of a longer name.
Variant: This would be another form of the name. It could be a foreign variant, a spelling variant, or perhaps another variant I can't think of. Jacques could be considered a variant of James and Kayleigh a variant of Kaylee--or the other way around.
Short form: Err...a short form of a longer name.
Pet form: A pet form would be a short form, heh. However, it has a connotation (to me, as least) of familial use and a certain unrelatedness to the bearer's actual name. As an example, I alone call my sister, Ruby, Vesta. Her friends don't. My parents and my other sister don't--but my other sister will sometimes call her Fetch. However, the unrelatedness to the actual name isn't really a big part, now that I think on it. Actually (sorry, I'm kind of going with the state-of-consciousness sort of posting style!), I think the idea could be that it's soft and very familial. But then, it could simply just be exactly like a nickname. It all depends on what you think.
Nickname: In modern usage, it's anything other than their proper name that you might regularly call another person. For example, I call my sister, Sara, Slinky and my other sister, Ruby, Boo. However, in the past, I think it'd relate specifically to names like Slinky--those that aren't related to the actual name. (Slinky is taken from one of Sara's favourite shirts, which has a slinky on it. Boo, on the other hand, comes from the fact that the sounds are in Ruby's name.) So things like Red and Curly would fall under it, while Sadie and Sallie wouldn't.
I think you can package short forms, diminutives, and pet names together--pick one of those words and use it. You could also possibly add nicknames into that. I'd leave the word derivitive alone, unless you say something like "Jamey, derived from James". Derivitive seems too general to me.
Feel free to correct me, because this is based off of what Merriam-Webster Online tells me, along with my own connotations dictate. I think that you're simply going to have to decide how you think each word should be used, see if your use can be backed up, and then explain your use.
If you ever want to ask me something, or just talk (because I think you're awesome, heh! I can't wait to hear more about this book!), feel free to email me at TaitoKalila at MSN dot COM.