Hm, I assume you mean Camtasia Studio 8?
Check the chart on the page you linked. You'll see that the version you have installed is in fact the latest version for Chrome. They haven't updated it beyond that. Each browser engine has its own latest version of flash. IE and Mozilla are 11.4.402.287.
Aside that, there's no reason Camtasia should be requiring you to have that version of flash. Make sure when you're publishing your content that you don't check off any options to "require the user to have the latest version of Flash". You should also (if possible in your project) try to target MP4/HTML5. Check any HTML being generated with it (hopefully using SWFObject) and check for the version checking code. There is absolutely nothing Camtasia needs to require the latest version to do. I've been using Camtasia (TechSmith codec mostly) for screen recording since version 2.