Why does groundhog day, work backwards from the way that I think it should (see details)?

So I hear that the legend goes, if the groundhog sees his shadow, he gets scared and runs in to his hole, for six more weeks of winter-like weather. And if he doesn't see his shadow, he stays up to play in an early spring.

An I correct in my understanding?

However, if he sees his shadow, I know for a fact that this is due to there being a mostly clear sky, thus lacking clouds. If he doesn't see his shadow, I know this is due to an overcast sky, filled with clouds.

In my opinion, a clear sky is a sign of early spring-like weather. And an overcast sky is a sign of six more weeks of winter-like weather. Probably the sign of more snow coming, or some other precipitation.