There are several UX /psychological concepts at play here which is probably why the pattern doesn't have a proper name.
The first part is called 'Gifting' - "Here's a FREE 7 day/2 week/3 month trial". This triggers the user's natural instinct to reciprocate somehow. This reciprocation could be just accepting the offer or simply holding the vendor in slightly higher regard than their competitors who don't offer anything for free.
Then comes 'Loss Aversion'. this is where the user has spent some time registering for and setting up an account that they feel more inclined to complete the purchase than to regard that effort as 'wasted' or 'lost'. Shopping sites use this all the time!
Once the trial is over we hit a whole bunch of concepts: 'Commitment & Consistency', 'Mere Exposure Effect', and 'Ownership Bias' as well as the 'Loss Aversion' from earlier. Our users have already committed to using the product and have the urge to remain consistent with their prior choice. They are already familiar with the way the product works and feel that they already part own it so why would they consider anything else?
UI devices that use one or two psychological principles to work tend to be named after or close to the principles they use. This one is quite complex and goes beyond a single page - there's a whole series of things going on within the UI and externally that make this pattern work.
You could just call it a 'trial period pattern/mechanic'.
Within your organisation, you can call it whatever you like as long as everyone involved knows exactly what you're talking about.
EDIT: One other thing to note.
The actual UI pattern here is just a simple 'Call To Action' or 'CTA' that simply asks the user to press a button - but the mechanic that is behind it changes the 'Action' that the user is being asked to take.