At the same time, you have to actually describe your profession somehow. Everyone solves problems. Solving problems is definition of productive work. A carpenter solves problems by cutting pieces of wood apart and attaching them together. A bricklayer solves problems by stacking bricks on top of each other, with a bit of mortar in between to smooth them out. A car mechanic solves problems by fixing cars. A surgeon solves problems by cutting people open and fixing their insides. Eventually it turns into a bunch of meaningless MBA platitudes. "We don't 'build houses', we 'deliver solutions' to people's shelter-related problems."
If you go to a good surgeon with a problem that isn't going to be fixed with surgery, the surgeon is going to recommend a different solution and possibly even refer you to a different kind of professional. "Hey, you just need to rest that knee and maybe get some physical therapy. Here's a physical therapist I recommend." People understand that surgeons are smart and will occasionally listen to them, so surgeons can get away with this. More importantly, surgeons understand this. Surgeons don't go around thinking, "man, I need to perform even more surgeries because I'm a surgeon and that's what surgeons do" (or, at least, they shouldn't)--they have a deep appreciation of what surgery is and when it is or is not appropriate.
So I don't mind thinking of myself as a programmer, because I'm a programmer the same way a surgeon is a surgeon. If you come to me with a problem that cannot be solved by programming, I will tell you that, because to do otherwise would be a form of malpractice.