I would just add that neither the requirement nor the authorization to employ "military force" against some party should automatically elevate the legal status of some situation to "war."

War comes with baggage that is best avoided, and is only triggered if the survival of the state is in question. Certainly the survival of the US state is not in question when it intervenes in the insurgency of another, so that is neither "war" nor "COIN," but is an intervention operation that could include a wide range of LOOs.

Similarly the State conduction COIN is typically not at "war" either, as the insurgent is as much a part of the state as the government is, but is engaged in illegal politics to compete for leadership of the state. It may get as violent as a war, but it is not war necessarily.

If, however, a third party conducts UW to incite and support said insurgency, then there may in fact be a state of war between the two states in question, but even that does not elevate the COIN operations against the supported nationalist insurgency to war. Wage war against the UW party, but conduct COIN as a civil emergency among one's own populace.

Out doctrine is a mess in this regards, and just dumps all of this inartfully into the "war" bucket, with predictably poor results.