Greg, youre pretty much correct. In the case of the W.A.A.F. (R.A.F.), the enlisted or 'other ranks' cap was made in wool to match the tunic (not wool serge which was thicker and rougher), and had a patent leather peak, while the officers version was in baretha wool with cloth covered peak and smaller crown. (Although W.A.A.F. Officers of Group Officer (equivalent to Group captain in the R.A.F.) and Air Commandant (equivalent to Air Commodore in the R.A.F.) had patent leather peaks on the cap with one and two rows of bullion oakleaves respectively, as per the male caps. There was no W.A.A.F. rank higher than Air Commandant).
The top of the officers cap also had a folded section that went from front to back and can just be seen on the photograph below (the cap on the left looking at the cap tops). In the W.A.A.F. the caps remained the same throughout the war, and up to February 1949 when the W.A.A.F. (Women's Auxiliary Air Force) became the W.R.A.F. (Women's Royal Air Force).
During the war, both the W.A.A.A.F. (R.A.A.F.) and W.D.'s (R.C.A.F.) changed their style of caps slightly, having started with caps based on the British W.A.A.F. style. The W.A.A.A.F. made only minor changes, but the W.D.'s brought in a totally different style. The R.N.Z.A.F. W.A.A.F.'s began by wearing a wide brimmed soft hat in blue felt, (similar to a panama) but by 1944 this was replaced by a blue wool beret.