In some ways the three forces approach makes sense as it organizes them by the source of these forces:
Thrust - A side-effect of an infernal contraption that converts fuel to noise
Weight - Indeed mass x gravity, the origins of which is therefore in the matter used to create the airplane
Resultant - The force that originates from hurtling a suitably shaped contraption through a medium called 'air'
Then obviously you can dissect the resultant into the bits we want: Lift, and the bits we don't want: Drag.
They are all still forces! Just because you apply a bit of vector mathematics to combine or split one or the other does not change the nature of the thingy.
I wouldn't go so far as call one or the other approach the 'correct' one. Both have their merits, the four forces approach is still the one that is most easily absorbed by us dim-witted pilots, so it makes sense to keep on using it. The three forces approach is in my view only relevant for the purely theoretical types.
I could even put forward a two-force explanation: Mechanically induced force and Aerodynamically induced force! The first one would be the sum of Thrust and Weight, while the second one is the Resultant as described above. The end result is equilibrium again, so it must be right!!
It's just a different way of looking at things...