What areas, if any, of graphic design could a person with color blindness be active in? Can a person who has problems differentiating colours actually work in such a field?
This is not an experts opinion, it's just based on my experience and thoughts.
When I read your question, I instantly thought of a fantasy painter called Ciruelo ( http://www.dac-editions.com/ - http://en.wikipedia.org/wiki/Ciruelo_Cabral). He is color blind and he has been working on illustration and design for years:
Ciruelo was born in Buenos Aires, Argentina, suffering from color blindness. That would not stop Ciruelo who at age 13 enrolled in Instituto Fernando Fader, an art school, where he began his formal studies. He started working as illustrator at age 18 for a graphic advertising company, and soon after began doing comic covers.
I imagine he had to find an equilibrium between what he thought was aesthetically pleasing and what other people did, or maybe in his case it was closely related. It might even be that the fact that he was color blind was what made him so famous: his color combinations were not expected or were outside the comfort zone.
You could start by doing some research on color blind artists and theory, and then maybe producing some work and show it to professionals and people in general (I'm sure everyone in here will be more than pleased to make a nice thread about it, and it could be very useful for other people) to see what they think. It could help you develop a "safe combinations" palette and reinforce what you already know.
Now, about he area.. I work in digital design so I'm not sure about print. I feel websites and apps (and interface in general) has more artistic freedom, but it's up to you, really, if you are determined I think you could work in any area.
Ignore the results of a test based on poor scans of printed cards shown on a screen, and go to an optometrist for a proper test if you are concerned.
I have some some colour differentiation issues. I worked as a graphic designer for quite a few years and I can't say it was ever an issue. But there's not going to be an absolute answer to this; there are many types and degrees of "colour-blindness".
My friend is an artist and he's color-blind. He has trained himself to be able to paint so that people who are not color-blind see it perfectly. So it may look funny to him, but to every one else it looks good. He's acquired that skill through just trial and error or painting and asking people on what to change to make it look more natural.
Don't let it limit you. If anything, use it as an advantage.
Being color-blind, you have the "advantage" of seeing things through the eyes of someone who is color-blind. So you can most certainly work in places that try to make their designs more accessible to different types of people. If that doesn't make sense: You can basically design color-blind friendly interfaces or advertisements (or anything else) because you are color-blind.
Fortunately your dream is not to become a fighter pilot!