Exactly. It just doesn't seem to be as weird/wrong if it's done that way. :-)

So with the Austen fanfiction, is it just a new story but with characters from some of Austen's books?

There are lots of different ones: New story with some Austen characters; Austen's stories from another perspective (say, Mr. Darcy's point of view versus Lizzy's as Pride and Prejudice is written); continuations; Austen's works in "modern times"; and "What if's" (e.g. What if Lydia had NOT gone to Bath? or What if Darcy's parents were still alive? etc.)