ISIL is not IS. FOX News is IN THE TANK.
Let’s just say the name of the group is up in the air. I actually think ISIL is more menacing than ISIS, because it means they want to take over the entire Mediterranean. And now they’ve moved on to IS, meaning they want to take over the world.
In any case, the use of the name in English probably doesn’t mean that much at the moment because it has morphed. They’re currently calling themselves the Islamic State (IS in English, EI in Spanish), however, so that should be the term.