Well, Cav, you DoD contact has it wired right - correlation of legal, policy and appropriations terminology. IIRC, a number of manuals (not sitting here in front of me) have been using that terminology for a long time. E.g., Operational Law Handbook 2007, has 36 hits on "contingency operations":

p.60
II. DOCTRINAL TYPES OF OPERATIONS

Military operations are divided into three major categories: 1) Major Operations and Campaigns; 2) Crisis Response and Limited Contingency Operations; and 3) Military Engagement, Security Cooperation, and Deterrence.[2] Joint Pub 3-07 further lists the following types of operations: Arms Control, Combating Terrorism, DoD Support to Counterdrug Operations, Enforcement of Sanctions/Maritime Intercept Operations, Enforcing Exclusion Zones, Ensuring Freedom of Navigation and Overflight, Humanitarian Assistance, Military Support to Civilian Authorities, Nation Assistance/Support to Counterinsurgency, Noncombatant Evacuation Operations, Peace Operations, Protection of Shipping, Recovery Operations, Show of Force Operations, Strikes and Raids, and Support to Insurgency.

Major Operations and Campaigns will most likely involve the triggering of Common Article Two. Other types of operations, however, will likely not. Of those, Peace Operations are the most common type of operation likely to involve large numbers of military forces, including JAs and Paralegals.
and

p.231
4. The Participating Nation Exception. As the JA proceeds through the regulatory flowchart of required analysis and actions, the most important and frequently-encountered problem is the “participating nation” determination. [27] This is because most overseas contingency operations do not generate the first, third, or fourth types of environmental events listed above. Accordingly, a premium is placed upon the interpretation of the second type of environmental event (i.e., major federal actions that significantly harm the environment of a foreign nation that is not involved in the action).
So, "overseas contingency operations" is a much broader term than say GWOT - of which more below.

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The Pentagon spokesman was pathetic - "I mean, I don't think a whole lot about it." So, off the top of his head: "...a campaign against extremists who wish to do us harm." Which now proves (tin foil hat wearers, listen up) that the Pentagon has successfully completed the long-awaited ESP project - and we now can positively identify our "extremist" enemies based upon what they "wish" to do to us. As the Chinese guy said: Know your enemy.

In terms of GWOT (which IMO cannot be defined either legally or from a national policy standpoint), the Pentagon guy could have said: We are engaged in an armed conflict with persons who are members of, or substantial supporters of, AQ, Taliban and associated groups. See AUMF re: Astan and 9/11.

Having said all of that, we must realize (in accord with our DHS secretary) that we are dealing with "human-caused disasters". So, instead of "terrorists", we should be calling our misguided brethren - "human-caused disaster implementers", or HUDIMPS. Yes, son, I fought in the HUDIMP War.