The ISIS is referred by many names, but the one they hate the most is Daesh. The ISIS will cut the tongue of the person who refers to the outfit as Daesh. They would ideally want to be referred to as the ISIS or the ISIL or just the Islamic State.
What matters for the ISIS is that word state has to be present in the name of the organization. This is largely because the ISIS wants to set up a government and does not want to be known as a terrorist group.
The hatred for the word Daesh:
The group has said repeatedly that any referring to the ISIS as Daesh will either be whipped or his tongue cut off. There is a lot of material that has been put up by the ISIS where it has spoken out against being referred to as Daesh.
Daesh is said to be a shorted version of the Arabic name of the group. The Arabic name of the ISIS is al-Dawla al-Islamiya fi al-Iraq wa al-Sham.
However the ISIS terms being referred to as Daesh as an insult. The world Daesh in Arabic could mean trample or bigot. However the word bigot is what the ISIS has a problem with and Daesh is more often than not means bigot who likes to control.
The French will call them only Daesh:
The Americans have often chosen to call the ISIS as the ISIL which has the word Levant at the end of it. The French feel that the ISIS is a misnomer and since the word state is part of the group's name, it gives a wrong impression.
The Islamic state would ideally imply that they are the governing authority of all Muslims in the world. This is something that the French want to avoid and hence they have decided to continue referring to the group as the Daesh cutthroats.
America on the other hand continues to refer to the group as the Islamic State of Iraq and the Levant or ISIL. Levant refers to the region around Syria which includes, Palestine, Lebanon, Israel and Jordon. Moreover Levant according to the Americans is a translation of the word al-Sham which forms a large part of the Middle East near the Mediterranean.
ISIS will settle only for a state:
The ISIS however continues to refer to itself as the ISIL or just the Islamic State. For the ISIS a name is everything and they will chop your tongue off if anyone asks them what's in a name? The ISIS has made it categorically clear that its agenda is to rule.
However looking at the pattern of the ISIS, it is clear that their intention is not terror strikes alone. They have been taking over the administration of places, implanting laws and even taking over oil wells to fund their administration. The manner in which they rope in recruits only indicate that they are building an army such as the one a nation would have.
Terror groups too have their own PR skills and when it comes to the ISIS, no one is better than them. Declaring a Caliphate and a Caliph was part of a long term plan which has in fact worked wonders for them. Further they would also want a name which has the word State in it as they feel it has a good ring to it.