The question “what kind of designer are you?” doesn’t quite make sense to me. When we look at the product a designer, artist, craftsman does, we evaluate the work based on the concept, aesthetic and functional qualities and not by the educational background or specialty, “label”, of the person. And from My point of view, people that accept or reject something based on the label of it’s maker, are having a narrow-minded view of the world, and not willing to break the molds!
If you look at the definition for “designer”, it's a person who designs any of a variety of things and implies the task of creating or of being creative. So narrowing it down to a single type of product or field is not the right way to approach it. They are just different skills and both take talent in different ways. I think that a collaboration of skills will give the individual designer a wider view and possibilities and these will be very valuable for him and the client. Anyone who communicates their concept through their product is a designer, independently from the medium used or the type of design you are trained in.