To cap or not to cap, that is the question here for our Indie-publishing authors.
This one is easy. If you can find the name on a map (for an actual geographic place/landmark) or on a birth certificate (for real people such as factual details for a biographical work), each are proper nouns/names and should be initial capped (each word starts with a capital letter). Same thing holds true for your fictional towns and made-up characters because your readers expect a character's name as well as that character's town/city and state to take that format.
Of course, shortening Elizabeth to Liz is commonplace and both require a capital letter at the beginning, because even though Elizabeth may be one lady's given name and thus shown on her birth certificate, so can Liz be someone else's actual given name and also typed that way on her particular birth certificate. A shortening of a name does not make it a nickname per se.
However, true nicknames, pet names, not a derivative of a first name, such as sugar, sweetheart, dumpling, honey, baby, sweet pea, etc., would all be as shown here with no initial caps. Just to be clear, I am only using italics for emphasis. When you use a nickname in your manuscript, no italics are necessary, unless you are being snide or snarky and want to show that to your readers.
Look up within Webster's "nickname" and then "proper name" which bounces immediately to "proper noun" for the distinctions discussed here. Your chosen style manual should confirm as well.
There are accepted exceptions to every rule, but just keep in mind what makes it clear to your reader. Like if your heroine's name actually is Sugar . . .