Good competitive skills means not just to compete hard, but to selectively compete, to harness your energy for the things that matter and, in some cases, to just realize it doesn't matter whether I win or lose here at all.
The important thing to know about playing to win and playing not to lose is that there are actually different neural networks that are being used. It's not very easy to do both at the same time and, if you are trying to have a playing to win mentality, you're going for it, there's some things that trip you up or trigger the wrong neural network. If you start worrying about your mistakes all of a sudden, if you get too focused on the facts and the details, these are going to shift your neural networks and sort of screw up your strategy.
I can push myself and you can push yourself, but competing, we push ourselves a little farther and we bring out our best even if one of us wins and one of us loses. The virtue of competition is that we both get better, not that one does. And that means we have great respect for the opponents, whether we win or whether we lose.