Edit 8/15/2014: It appears this has been fixed in the latest Steam patch.
In Windows 8/8.1 64-bit, I know that Flash is built in to Internet Explorer 11 so you don't have to download it separately. And then I installed Steam, and videos in the Steam store worked fine so I guess Steam knew how to hook into the Flash player that was built in to IE 11... but then I installed Firefox, because there are a few websites that just don't work in IE and plus Firefox has a nice built in PDF reader, but I still use IE 11 as my primary web browser. Now Steam videos don't work any more. They tell me that I should go install the Flash plugin.
How can I make Steam go back to using the IE11-based Flash player that it was using before I installed Firefox? I don't want to go install a Flash plugin for Firefox when Steam videos were working fine before I installed Firefox in the first place.