Chef or cook?

I was just wondering: what is the difference between a chef and a cook?

To me, a chef is someone who has had professional training and worked his/her way up through the ranks of a professional kitchen, from prep cook, to line cook and sous chef. Also, I think a chef is innovative, not just coming up with new dishes, but whole new ways to look at and approach food. That means just as much concern with presentation as taste. Most cooks, I think, are not very concerned with how a dish looks as long as it tastes good.

What do you think?