GRpr
You're right, in Canada you can get away without filing an IFR flight plan if you won't be entering controlled airspace; you can file a flight itinerary (e.g. tell your mother) instead. And yes, the primary purpose is SAR.
Are the planless flights in IMC in the UK actually IFR flights? They do have a separate rating there that allows pilots to fly in IMC without having an instrument rating.
Your suggested definition is well-written, but as you say circular, and it becomes even more circular when you use it to expand any instance of "IFR flight" in the air law: you end up with something to the effect of 'flights that are required to follow the rules that such flights must follow are obliged to follow this rule'.
PL77
Regarding IFR navigational instrument requirements. When I initially did my instrument rating the memorized answer was "two VORs, a VOR and an NDB, or two NDBs, plus you must have at least one NDB in Northern Domestic Airspace." When I was working on my IATRA I searched and searched for the source of this in air law. All I could find was CARs 605.18 (j). My course instructor considered that definitive.
605.18 No person shall conduct a take-off in a power-driven aircraft for the purpose of IFR flight unless it is equipped with
[a-i are the equipment required for VFR flight, and equipment unrelated to navigation]
(j) sufficient radio navigation equipment to permit the pilot, in the event of the failure at any stage of the flight of any item of that equipment, including any associated flight instrument display,
(i) to proceed to the destination aerodrome or proceed to another aerodrome that is suitable for landing, and
(ii) where the aircraft is operated in IMC, to complete an instrument approach and, if necessary, conduct a missed approach procedure.
So if your errant military pilots are flying in constant radar contact, with a PAR approach available at the other end, and they have two comm radios and a transponder, they may be able to legally claim that they have all they need.