The distintive line is when the level of violence exceeds the capacity of the civil governance, requiring them to bring in the military (much like our MSCA construct for all other forms of civil emergencies) as last in, first out, and always under civil control as a crucial augmentation. I have posted a chart that shows this construct. Key is that it recognizes that the goal is both to reduce violence AND improves the aspects of poor governance giving rise to the insurgency.

As to what we do when we deploy, that is like going to the neighbor's house to help him with a domestic dispute. Start thinking of it as your problem to solve, and you are in for a world of bad higher order effects. We'd never do it in our neighborhoods at home, yet we do it every time when we go to our global neighbors as a country. Problem is because we think of these things as "foreign wars" and not "foreign domestic/civil disputes."

LTC Brian Petit's article on "Think COIN, but do FID" is a great supplement to this line of thinking. links to it are posted here on the SWJ.