Your DD sounds just like mine.
Her DD sounds like me, except I'm not a gymnast
I love competing, and I love learning new things and doing well and working hard and all that- I'm not a slacker. And yes, I like to win. But I don't freak out about losing. And I think part of me expects to lose, so that when I lose it's like "Ok, I just lost, whatever, back to work." And when I win, it's icing on the cake. In sports, in academics, in my career. The only time when losing pisses me off is when I know I'm the best, and I just screwed up. That happens sometimes.
So if I think about myself now and as a child, I don't think it would have ever been possible to light a fire in my belly. And I think most people like me probably won't really be changed by another person. You can get a person to want to win something specific, you can motivate them for specific tasks or competitions, but not to have competitive drive that comes from within. I remember a specific thing.. when I was a camp counselor back when I was 19 there was a doubles tennis tournament. I was paired with a friend, a counselor from Mexico. She wasn't a very good tennis player. I was pretty good- not great, but yes I could play. We had to play against a similarly uneven team- the better player was much better than me (or so I thought- she was definitely fitter), and the worse player was pretty lousy. We probably would have lost, except after the first game my teammate looked me in the eye, smiled intensely, and said very seriously in her super cute Mexican accent "I Want To Win." I was like, uh.. ok, then I guess we'll just have to win. And win we did, and I carried us. But if she hadn't said that, and if I didn't really like her and want to do it for her (because she wanted it so badly), maybe we wouldn't have. It was an interesting lesson for me.
Since then, I'm really only motivated to win by external factors... if I sense a rivalry with my opponent, I want to win. When I didn't make a particular team, so I joined a different team in the same city, I HAD to beat the boats from the team that turned me down. Or at least I had to really die trying. But if it was some unknown opponent.. eh, whatever, let's try.