The group should now be referred to the literal translation of الدولة الإسلامية: Islamic State. The organization has established a political theocratic state based on the Mohammad-era war Caliphate model. It also now claims worldwide control of all Muslims. It no longer considers itself limited to Iraq and the Levant.
Just as we recognized the name change from al Qaeda in Iraq to the Islamic State of Iraq to the Islamic State of Iraq and the Levant, we should recognize this name change. No more AQI. No more ISI. No more ISIL/ISIS. We are at war with the Islamic State, the self-proclaimed Caliphate. If we do not consider their religious angle seriously we will fail in countering them on a cultural level.