The fact that it requires “your library” and not “a library”, and that “Magicians and rangers are assumed to add new spells they
have learned to their libraries whenever they visit them”, suggested to me that the library must contain a spell for the caster to be able to scribe it to the spell book. Since you automatically update your library it probably won’t matter much, but if you had access to another casters library it could make a difference whether you are able to update your spellbooks or not.