Look at the definition again.
"...ability to project military force in air or space by or from a platform or missile operating above the surface of the earth"
BH, in general, provide the ability to project military force in the land environment - not in air or space. It's all about effect - where is the effect being generated? What effect in the air environment is a Chinook generating? It only uses it to deliver troops from A to B faster than a Bedford. You could argue, therefore, that CAS is also part of the land environment, for that is where its effect takes place.
To me, air power is about effect in the air and space environment. DCA/OCA/AAR/AT/AI/strat ISTAR are all therefore part of the air component. Just because it flies does not make it airpower - which is why Land/JHC own BH, and HQNI owned heli teli in NI.