Please go to below link and click on check now button:
Check if it is installed.
Also Right click on youtube video and check if it shows Adobe flash player version, as it may be possible the video doesn't play using flash player.
If I go to the link you mentioned, my computer tells me that flash player is installed (conflicting information from the following link - Uninstall Flash Player for Mac OS).
Although it's great to hear Adobe say that I have a flash player installed on the link you added Kratika, Adobe Flash is still not working on youtube videos that use flash player. It seems that this is a common problem for other OS X 10.6 users, as I have seen similar issues on multiple forums. Helping me could potentially help you guys receive fewer questions...
Thank you for a very detailed explanation of the issue. It was very helpful. The last screenshot you posted, showing the System Information, indicates that Flash Player is indeed installed. If it weren't, the installed Flash Player version would not be listed.
I went to that YouTube video and it's not being delivered via Flash Player. It is being delivered via HTML5:
Do you have examples of other Flash content that is not displaying as expected? If so, please provide the link to the content.