i had the same problem that you had for a long time until i found this forum Vincent. Bit by bit I have collected some clues and I hope its not too late to share my research with you. Heres how I fixed thanks to Carl E. Myers that gave me the answers to identify the problem.
The way I discovered the problem is when I saw the control panel/flashplayer that Carl E mentioned, the Flash ActiveX and the plugin were not installed as 32bit in which I will explain later. Based on what you post, it is most likely your problem as well.
so here's how to fix it:
http://www.adobe.com/support/flashplayer/downloads.html
1. On this page you will see
Download the Windows Flash Player 11.2 ActiveX control content debugger (for IE) (EXE, 3.93MB)
download, and install
2. Same page, you will see
Download the Windows Flash Player 11.2 Plugin content debugger (for Netscape-compatible browsers)
That should do it. Why?
Adobe Flash Installation installed it as a 64bit!!
Your computer is a 64bit, like mine, so that's why ie and firefox are not picking up our installation of Adobe Flash because Adobe Flash automatically installs to the windows system you are running. This is a big problem us 64bit users because some application are NOT 64bit but more so compatible (if that makes sense). For example, Mozilla's Waterfox IS a 64bit application that is built for our windows and you would see that the flash would work with no problem because the flash was installed as a 64bit. Also, IE64bit, another evidence that should be to able as well, but don't get confused with IE as it is a 32bit application.
The activex and plugins you installed are manual 32bit installation and which you should now understand why it will now work for firefox, ie, and other 32bit browsers that needed it. Apples to apples now.
P.S. Even though that should fix it, I highly recommend using Waterfox instead of Firefox as it is amazingly faster and way better experience for your windows.