Ah, I see. Ya, the cPanel email provided requires nothing extra in Thunderbird. "My site's" email was vague enough for me not to understand, but whatever, you got it working.
If you have any questions, since it appears you're new to Thunderbird, feel free to ask. You'll find that it's a great email client. The cool thing with Thunderbird, along with Firefox, is the various amounts of extensions which you can use to customize to your liking. Enigmail is an extension for Thunderbird which integrates the OpenPGP encryption standard.