In this case: OS is Debian 6.0.4, 2.6.32-5 64 bit kernel, Iceweasel (Firefox 3.5.16 unbranded) browser.
These appear to be identified correctly, and the offered version is 188.8.131.52, which I assume is correct.
The browser does not provide for the "apt" protocol, so using the Ubuntu version is not an option.
The YUM and .rpm options are unsuitable for Debian and Debian based distributions.
The .tar.gz file contains no text instructions and no install script. Earlier versions I have used included both.
Paths are not absolute, so it is unclear where libflashplayer.so should be installed.
I note that Debian provides a package, flashplugin-nonfree, which appeared to install the appropriate package. It is possible, even likely, that this package will operate correctly on many of the other Debian based distributions, of which there are quite a few. Please consider mentioning this. However, in any case, please provide instructions to go with the tarball for users who must use it to install flash.
Modifying the observation about the Debian flashplugin-nonfree package: It installed libflashplayer.so in /usr/lib/flashplugin-nonfree/libflashplayer.so, but left an old libflashplayer.so (possibly from an old install done with the Adobe flashplayer installer) in /usr/lib/iceweasel/plugins/. With the resulting configuration, flashplayer in FF 3.5.16 continued to demand upgrade for some sites. It works fine if the existing libflashplayer.so is replaced by a symbolic link to the new on in /usr/lib/flashplugin-nonfree/ or removed completely. It is not clear whether this is a fault of the Adobe Flash Player packaging , the Debian flashplugin-nonfree package, or a characteristic of my particular installation. I have reported it to the Debian maintainers.