If it is giving you a direct link then it will probably be to the latest release version, and this will not have debugging capabilities. Rather than using the link you can download the debug version of the latest flash player here:
Not sure which one to download.
Adobe Flash Player 10 — Debugger Versions (aka debug players or content debuggers)
Adobe Flash Player 9 — Debugger Versions (aka debug players or content debuggers) for Flex and Flash Developers
Since Flash Player 9 explicitly says "for Flex and Flash Developers" and version 10 does not, I am inclined to believe I should use version 9. Is that correct?
All versions of the debug player are for developers of Flash / Flex content. I would recommend having the debug version of the latest player (version 10) installed. This should have all of the latest security fixes.
If you want your application to work with Flash Player 9 then you might choose to install that instead, or rollback to it during testing.