"fault" is a mistake with responsibility, so it depends on if you feel responsible or not. For example, you might say: I made a big mistake picking that restaurant but it wasn't my fault the food was so bad. It's been great every other times I've been. or It was my fault dinner was so bad last night. I should have found us a better place to eat.

I agree. 'Mistake' is just an error of judgement. 'Fault' is taking the blame.

I would say, "I made a mistake..." as the start of an explaination of what happened or to offer an excuse for my behavior. You would say, "it is my fault" when there may be a question of who made the mistake and you want to take the blame and move on. If you are playing volleyball, and you get in the way of your team-mate and lose the point, you say "That was my fault" and that lets everyone know that you are correcting yourself. To say "That's my bad" is new and hip and very light-hearted (and yes... American).