The answer found on this server fault thread is my jumping off point for this general topic: https://serverfault.com/a/313558
... and this question can be thought of as a follow-up to that answer.
Does my SSL cert have anything to do with or say about the symmetric session key? (I know that the browser and web server both have a role there, but what about the cert itself?)
If YES: Where is it specified in the cert?
If NO: Why do CAs all boast about "128-bit to 256-bit encryption" which is implicitly referring to the session key?