The Blank Slate: Designer or Illustrator?

I once heard someone describe the difference between a "Designer" and an "Illustrator" as follows... A designer assembles pre-exisiting elements such as photos, vectors, and textures to create a final piece. Whereas an Illustrator creates a piece by hand, creating element from scratch using his/her imagination. While, I believe that there is some truth to this statement, I don't think it's that cut and dry.  I believe that designers and illustrators are like engineers, and often times the line between the two is blurred. The more resourceful an artist is, whether it be collecting resources or pre-existing elements and effectively applying them, or putting a pencil to paper and assembling lines and shading to create imagery from scratch, it's still a matter of effectively conveying the image or scene that the artist sees in his head. In my opinion, that's what makes a designer or an illustrator an artist. This brings up the question...

Do You Consider Yourself a Designer or an Illustrator, and Why?