Hi Dominic,
1. I did delete and reinstall the Flash Player.
2. "Allow plug-ins" was already enabled.
3. I didn't remove all website data because I needed some of the data.
but, what I forgot to mention in my first post was that I also had ClickToFlash.
It turned out to be misconfigured. I changed these:
ticked ClickToFlash > General > Load Flash if HTML5 conversion fails
ticked ClickToFlash > General > Use HTML5 media fallback
un-ticked ClickToFlash > Media Player > Default media player > HTML5 > Load media player automatically
ticked ClickToFlash > Media Player > Default media player > HTML5 > Show preview image
selected ClickToFlash > Media Player > Nonnative formats policy > Use only as a last resort (instead of "Never use as default")
ticked ClickToFlash > Media Player > Initial behavior > Instant autoplay
ticked ClickToFlash > Media Player > Source Selector > include media sources
ticked ClickToFlash > Media Player > Source Selector > Include Flash
By reinstalling the Flash Player and changing some ClickToFlash settings, I fixed playing flash videos on the sites I couldn't before.
Many thanks for your help.