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> Android is that OS itself and just the OS.

Actually, Android is a Google trademark and you have to do what Google says if you want to use it.

Amazon uses AOSP but it can't call it Android.

This makes sense. Anyone could take AOSP and make it incompatible with Android (TM). If they could call it Android, that would mislead consumers.

The problem is that if you ship a single Android phone with Play etc, you are not allowed to ship any phones based on AOSP. In fact, Google just made Acer cancel the launch of an AOSP phone aimed at the Chinese market because it is already shipping genuine Android (TM).

This is not "fairly unrestricted" ;-)