For me one must apply the flagpole test.

Go the capital of the place you are at to deal with an insurgency, find the tallest flagpole and look to the top of that pole. If it is the flag of your nation, you are conducting COIN, but if is the flag of anyone else, you are not.

COIN is not unlike parenting. If you are at home dealing with your own children you are parenting. But if you go to a neighbor's home to help them with their children you are not parenting. To think you were parenting would lead to a disaster of undermining their role as parents and regardless of how good your intentions or actions, would probably not leave a functional family behind when you leave to go home.

We understand this intuitively in operations of the scale of a family - but for some reasons lose all common sense when we scale it to the national level, rationalize our actions with interests or threats, and label it "war."

But it is the same damn thing.