I have always done a little of both, but I definitely would say that I was a snowboarder first. I never took skiing seriously until shaped skis were around. Once a ski could carve well, I finally thought that the fun rivaled snowboarding. Now I enjoy skiing quite a bit.
I got my AASI Level III before I did anything on skis. I have now received my PSIA Level II, but I think that is as far as I will go. There is just no diversity in PSIA teaching and skiing IMO. The public is going more toward twin tips and freestyle skiing and PSIA is stuck in their "Centerline model".
In a snowboard exam, levels I - III, you will be in the park and pipe. In ski exams and clinics, I have yet to go into a park or pipe. I even attended a park and pipe clinic that never went in!!! At the end of the day, the comment was "you do not have to be able to do it to teach it" That is a clever way to say "I can't do it".
Oh, sorry for the rant - back on topic.
I feel that snowboarding has helped my skiing tremendously. When I finally got on skis (after a few years off) in the late '90s, I just hopped on and thought about what I do to make my snowboard work. I transferred that to the skis and layed sick arcs immediately. I know a lot of other snowboarders who have done the same thing.
In the end, I believe that a truly good snowboarder can transfer things easily. I would also say that a truly good skier can transfer things to a snowboard easily (of course there are exceptions to every rule).
BTW.- here are my faves: