"Woodwind instruments are made of wood"
These tools can also be made of metal (e.g. saxophone or flute). Alpine wood horn is not a wood instrument but a brass instrument group. Also, the group includes all the tools through which air passes very quickly from the mouthpiece to the pavilion. If the airflow is already dispersed when blowing or diverted then it is a woodwind instrument. Even a comb can be considered such an instrument if you blow on it. Subdivision is somehow curious since the first woodwind instruments were whistles, the first brass instrument group were copies of animal horns, which could easily blow.