Streaming rental movies on Itunes

You don’t have to download the full movie in order to stream (when you have a good Wi-Fi connection).

Your device will want to cache (download) small snippets of content though, to ensure smooth playback without buffering pauses.

Your device should have some free space to operate efficiently (2–5 GB or more). Almost full disks have no room for the system software to operate with temporary files of all kinds, including buffering.

That seems to be a misunderstanding.

You can rent or buy from Apple’s iTunes Store without subscribing to anything. The optional subscription from Apple is called Apple TV+, and features Apple Originals exclusively (not available elsewhere). Renting, buying or subscribing from Apple will be billed to your Apple ID account.

You can rent or buy from Amazon without subscribing to Prime. The Prime subscription from Amazon will give you perks for both shopping with Amazon (tangible goods), as well as selected movies and TV shows that are included with the subscription. Renting, buying or subscribing from Amazon will be billed to your Amazon account.

Subscription content may not be transferable through Movies Anywhere.

If that's the case then there would be further investigation needed.

The Apple TV app is free to use, the subscriptions are only required if you want to access content within Apple TV Plus or one of the other Apple TV channels that are hosted within that app.

Amazon Channels work differently, in that one cannot subscribe to those unless they already subscribe to Prime Video. If one just wanted to do a rental or purchase then that is completely separate and has no tie to a subscription.