I think that it is dangerous to resume war to just 3 kinds of wars, based on one military adventure and analyzed from only one side.
ICRC came with international conflicts and non international conflicts and that is already complex enough.
I will use the example of Lebanon 2006: a war between a State (Israel) and a militia group (Hezbollah) that took place on the land of a third party: Lebanon State.
Is that an intra state war? Is that a counterinsurgency? Is that a Civil war?
It is already difficult to know if it is an international conflict…
Spanish civil war in the 30th. A group of rebels (the fascists) used the State tool (the regular army) to over come a legal government (the partisans) organized in small armed groups to defend the legitimacy of the law.
If we call it the Spanish civil war, it is because it is a non international conflict between the people of one country and not the consequences of an external intervention, as is and was Iraq.

Wars are more complex than just 2 or 3 boxes. There are means, aim, objectives…
Iraq is a model for counterinsurgencies, yes. And should be analyzed to understand how it is or not part of a regular stabilization operation.