Word for changing or shortening or abbreviating a given name

hypocorism: (also known as a pet name or calling name), is a shorter or diminutive form of a word or given name, for example, when used in more intimate situations as a nickname or term of endearment.

James --> Jimmy
Charles --> Charlie

You are probably looking for a word diminutive which is a noun:

A diminutive word is a "cute" version of a regular word, "Maggie" is the diminutive of "Margaret." ^{per vocabulary.com}

So your "have XXXXXXXXXXX her formal name" = "have formed a diminutive of her formal name."

You named it yourself, nickname is the term you're looking for.

nickname

(n.) a familiar form of a proper name, as Jim for James and Peg for Margaret Random House

(v.) to give a nickname to American Heritage® Dictionary