First of all, sorry to hear about the laptop failure.
Now, when you install OneCare, it is always in trial mode until your subscribe/activate.
Open OneCare, and there should be a link near the top that leads you to subscribe or activate a subscription to OneCare.
If that link is not there, check your PC date and time. If it is a future date, the link will not appear and OneCare may report that your trial has expired.
I do believe that the wording may be incorrect on the members page:
When you do get the link (which may actually say "subscribe"), you will be prompted to sign in with a LiveID. Per number 3 above, use the same Live ID you used previously.
If you continue to have problems, please contact support. If the other PC still has OneCare activated on it, you may want to follow the path explained in the FAQ post to get to support on that PC since the new laptop will be in trial mode and will require that you get support via email.