The trial period is in effect unless you also reinstalled OneCare after changing the PC name. I don't understand why you say that you rebooted and signed in with the LiveID for both PCs or what you mean by "add the newly named laptop." Downloading the trial does not require signing in, though it suggests that you do this to try out the multi-PC functionality. If you signed in on the laptop already, before you changed the PC name, it is already part of your Circle, though is may be incorrectly identified.
So, please clarify the order of events, what your status is on both PCs with OneCare. I'm also curious as to the reason for renaming the PC.
-steve