*Support but change disambiguation - If I can interject, I'd like to add that I can see good arguments raised by both sides in this discussion from what I have read so far. 

The current title used has a natural disambiguation, which is always preferable in an article name and obeys WP:TITLE and WP:NATURAL. I don't wish to make a statement without any evidence to support it, but where I come from, everyone knows what the term 'Islamic State of Iraq and the Levant' means and what it represents so it's not unpopular. The acronym 'ISIL' is also accurate enough to represent the group. That being said, I have started to notice that both these terms are being displaced by 'ISIS', 'Islamic State' and 'IS' in publications that I read, and news that I tune in to. Many reliable sources are also starting to refer to it as 'Islamic State' with many many different variants. The other fact that also should be considered here is the actual wording in the name. 'Islamic State of Iraq and the Levant' makes the group sound as if it is geographically locked in these two regions, with no territory or ambition to spread outside of its primary spheres of influence. Similar to the Taliban-controlled Islamic Emirate of Afghanistan. And, dare I say, it also makes them sound more legitimate as a group then just the ambiguous 'Islamic State' designation, due to the emphasis placed upon the areas of 'Iraq and the Levant'. Makes it sound more formal. But it is now not the case. As multiple editors have stated, it is not merely a parochial militant group with territory in the regions of Iraq, Syria and now even parts of Lebanon but a transnational entity bent on expansion. I cannot see how relegating it to two geographic positions is the mature thing to do in an Encyclopedia, whatever the intention. It is becuase of this that I do not believe the current name and its disambiguation is viable anymore, purely for this reason. Some editors had stated the potential and very real problem of justifying 'Islamic State of Iraq and the Levant' territory outside of Iraq and the Levant. It's not a recognized entity as far as I know. You cannot liken it to the Russian Federation having territory outside of Russia in Europe e.g Kaliningrad or the United States having territory outside of North America e.g Hawaii. What I am getting at is that this implies a level of legitimate ownership, and this group doesn't have it. Not in Libya or Yemen or Iraq or Syria or anywhere.

The issue I have with "Islamic State (IS)" is the disambiguation which, whilst being a widely used acronym for the group, can not be measured accurately to satisfy WP:COMMONAME through Google searches alone. Even though many sources use 'IS' which has been stated, on its own as a disambiguation, it is simply too generic. A similar issue arises with the term "Islamic State". However, were you to isolate the period for a search, the results would become more accurate, and this has already been stated. If a more appropriate disambiguation were to be found that could in concise terms explain what the article is about, I would support it. I suggest something along the lines of Islamic State (organisation) or Islamic State (ISIS) or Islamic State (group) or Islamic State (militant group). I am under the impression that WP:OFFICIAL and WP:COMMONAME are the important policies here, as editor  stated, and in my opinion the current title doesn't satisfy either, whilst a proposed title has the potential to satisfy both. 

The only reason why I choose to support this current proposal, is that progress towards a better article name will emerge allot quicker were it 'Islamic State (IS)' vs. a proposed alternative rather than the current title vs. a proposed alternative.