There is a class of noun called, interestingly, Picture Nouns. These include picture, description, story, painting, and any other noun that refers to a representation of something else. There are hundreds, and they have very peculiar syntax, because they're very peculiar semantically. All nouns are representations of something else, but picture nouns are representations of representations.
That means (using the Mind is a Container metaphor theme), that picture nouns have content, i.e, whatever it is that the picture, the description, the story, the painting, etc. represent. And that content may be referred to with the practically meaningless preposition of.
So my picture may mean
- a picture that I own
- a picture that I made
- a picture that shows me
as well as many other things, not restricted to picture nouns, like
- a picture that I sold/bought/signed/
- a picture that I am particularly fond of
- a picture that I mentioned in a previous utterance
Whereas a picture of Bill can only mean
- a picture that shows Bill
while a picture of Bill's may mean the same thing, but may also mean
- a picture that Bill owns
- a picture that Bill made
Summary: Possessives do not always refer to ownership;
or, perhaps, ownership has more dimensions than one might expect.