I wouldn't really consider small wars per se to be part of LIC, but rather the reverse.
One reason I tend to prefer the term "small wars" is that it often reflects the overall level of a nation's commitment in terms of time and treasure to the conflict. Notice that I use "nation" in preference to "military" because that's the level of policy that is being discussed, IMO. It's still engaging all aspects of a nation's power, just at a lower (smaller) level than a full-scale war might. To me the term LIC is just too limiting, as it implies just the military side of things and tends to hint that somehow it isn't a war...which again to me sends up signals of "disposable" forces and objectives.
The commitment of Marines in Central America during the 1920s and the British colonial involvements over the years were questions of national policy and engaged many elements of both nations' political and military infrastructure. They were certainly wars (although with limited forces - the objectives weren't always as limited), but the overall commitment of time and treasure in comparison to other national activities was small.
I would hazard a guess that one reason the US has been so anxious to create new terms for this (LIC, MOOTW, alphabet soup....) has more to do with domestic political considerations (possibly including the fact that the use of the term "small war" might get some in Congress excited about needing a declaration of war) than it does actual utility. The resistance of some of the services to working with the State Department (and vice versa) may also play a role.
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