Your User Agent says you're using - Mozilla/5.0 (Windows; U; Windows NT 6.1; en-US; rv:1.9.2.3) Gecko/20100401 Firefox/3.6.3

Your UserAgent string in Firefox is messed up by another program that you installed.
http://en.wikipedia.org/wiki/User_Agent

type about:config in the URL bar and hit Enter
If you see the warning, you can confirm that you want to access that page.
Filter = general.useragent.
Right-click the preferences that are bold, one line at a time, and select Reset.
Then restart Firefox

I tried that, per your instructions, but it doesn't seem to make any difference. Still getting the same message on my Firefox home/start page, asking me to upgrade to the latest version, when I already have done that. Could this have anything to do with any of the Firefox add-ons I have? I was thinking of deleting them all just to see, but was interested to know, in advance if possible, if this will make any difference?

Even if your useragent was correct at Firefox 8.0.1 that old Firefox homepage hosted by Google would still give you that message.

The www.google.com/firefox/ was a old default homepage used before Firefox 4.0 and is not maintained by Google anymore. Either ignore that message or change your homepage to about:home if you want similar one or some other page or leave blank for no homepage.

Tools->Options->General->Homepage: where you can change it.

OK, I see what you mean. I can easily change it from the obsolete google/firefox homepage, or leave it if I don't mind the silly message. Thanks for clearing that up.