That message is on the Moto X. If you had wanted to move from your current 1.0 “unlimited” plan to the 2.0 refund plan, you would have to wait until your billing cycle date to do so.
Using the Moto X, please cancel that pending transition. Is there an option to do so on that same page you just quoted?
Then, yes, using the same account username and password that you use on the Moto X, please follow the steps in the activation guide.
If the phone has multiple Google Accounts, it may be that you’ve synced apps, photos, and contacts to different Google accounts. So if you get the new phone set up and find something is missing, you may want to check the old phone to see what account that missing something was associated with.
The content on the old phone will not go away when it is de-activated. It will stop making calls and sending text messages, but apps will keep working, as they would on a tablet, on Wi-Fi.