In art school one of my instructors asked the class an interesting question: Are you an illustrator who designs or a designer who illustrates? It was an illustration class but our assignment was to design a personal logo with a matching set of stationary. Our instructor was an illustrator who had been paid to design a few brochures.
At the time I was an illustration major. I had been a graphic design major the first semester but quickly switched. So of course my answer was an Illustrator who designed. Being a graphic designer was considered a fallback for those who wanted to be creative and create art but couldn't.
The funny thing was I really enjoyed my graphic design class (the only one I took until my MFA almost 15 years later), and I was always sneaking into the graphic design classes to observe instruction, critiques, and presentations. Just a few years later I accidentally became a graphic designer/web designer and have been doing it for the past decade.