I think we're missing the point. ISIS and ISIL are both wrong. ISIS/ISIL was a group of people trying to establish an Islamist caliphate. They have successfully been running a country for the past few months, with their own courts, their own "police," and their own funding. I think they succeeded in that goal (at least for now), so they are no longer just a group of people trying to establish a caliphate. It's more correct to call them the Islamic State. They're not just the terrorist group ISIS/ISIL anymore.
Not that this really even matters much. Edit: I take that back. What we call "our" enemy actually matters a lot. It's just that when media figures refuse to use the term "Islamic State," or when they say "so-called" Islamic State, it really exemplifies how in denial they are that their precious war wasn't successful.