One thing I've noticed is that Firefox insisted on me downloading the very latest version of Flash before it would play embedded video, wheras IE was happy with the older version.
I think you are wrong.
I.E. uses a Flash thing which incorporates Active X and all its security vulnerabilities.
FireFox is much safer, but you have to use Firefox to download/instal the safe version.
The two executables are totally independent.
I.E. cannot use the safe version of Flash.
Firefox cannot use the dangerous version.
Probability is that Firefox MAY have found it had a much more obsolete version that I.E. was using - perhaps there was no version at all.
You have to visit Adobe Flash with the browser for which you want Flash.
Alan