No, it does not really depend on the definitions. Proudhon's idea of private property being theft is pretty much the same kind of lunacy that is a nail in the coffin of any dyed in the red anarchist. This while most anarchists make an arbitrary distinction between private property and personal property. So you're allowed to have stuff, you're just not allowed to own the capital goods that produces that stuff, nor are you allowed to own a place to keep everything.