To me, a hacker is someone who thinks creatively about problems, and who views technology as a set of tools to solve them. The YC question about "the time you most successfully hacked some (non-computer) system to your advantage" is, to me, most resonant of the 'hacker' ethos. Although many answers I've seen to this question involve using a computer system to solve a non-computer problem, the sheer variety of non-tech "hacks" out there speaks volumes about what a 'hacker' is and does.
To me, the label 'hacker' isn't claimed, it's earned. I call myself a hacker, though I didn't until someone else did. I've never broken into secure systems armed only with a 28.8 bps modem and active matrix screen, though -- I use it entirely in the 'problem-solver' sense of the word, and proudly.
Having programmed since childhood doesn't make me a hacker. I owe far more to years of Latin study, an unhealthy interest in logic problems and strategy games, and being trained via school that there is always a solution to every problem if you apply yourself hard enough.
But on the other hand, if I were just learning to program now, how much of the 'hacker' mindset would come along with it? Very little, in and of itself; I tutor beginning students and I'm always trying to teach them how to solve problems and look for answers, how to be creative and elegant, and how to reuse other people's work -- it wasn't until I started mentoring that I realised how little of this some people do naturally, and I still don't know the reasons why that is.
On a side note, and having just re-read Little Women, I have a thought about hacking and poverty. When you are poor, you have to be creative. How much of that mindset overlaps with what makes a good hacker? How many hackers grew up in disadvantaged circumstances and learned to make the most of the resources they had? How many hackers hung out in libraries, absorbing information like sponges, because it was free and warm and both their parents were at work?