As I've always understood it, the idea comes from the cross product and angular momentum. If we can agree on an 'up' direction (the positive z axis in a coordinate plane), then we say a rotation is positive if the direction of its angular momentum, as determined by the right hand rule, is upward (that is, has any positive z component). This does end up agreeing with the winding number argument - counterclockwise in the xy plane is positive.
As I type this, it makes me wonder - what if it's perpendicular to the z axis? Well, I don't know. I suppose I'm out of luck. But such strange times might call for strange measures, and we could actually describe the angular momentum vector to remove all doubt - or we could just change coordinates so that 'up' isn't quite how we thought of it before.