(a) There are lots of regional signs, so signs in one area of the country may be different than signs in another. The book may have been published in a different region of the country.
(b) All languages change, ASL included. (Ex: 15 years ago, what was a 'mouse'? Nothing to do with computers, that's for sure). The book may be old.
(c) Signs do not follow the english word, but rather the meaning of the word. If you look up the sign for "break" in a book they may give you one, POSSIBLY two ways to sign it, but I can think of at least 5 that I use in different situations. They wouldn't all be included in a book, but using a different one is wrong in ASL.
(d) The pictures in ASL books are hard to understand, and your mother-in-law may not be doing the sign correctly.

Either way... your daughter is 6. Even if she was making it up, if it's making her happy, who cares? But if she is learning it from a deaf girl, and an interpreter, I'd trust her over a book.