ActiveX Filtering In Internet Explorer 9 and 10 Kills Flash Player

I recently upgraded to Internet Explorer 10 and promptly discovered something odd - Flash Player seemed to be broken. Whenever I tried to access a Flash application or video, it would tell me that an Adobe Flash Player upgrade was required. Take a look at these examples...

At first, I thought that perhaps the new Internet Explorer 10 required a newer Flash Player, so I downloaded and installed Flash Player - nothing happened. I restarted Internet Explorer 10, and tried again - it still asked me to upgrade Flash Player!