Both are just a modified Linux.

They're distinct ecosystems though because Android TV/Chromecast use the Play Store for content/app delivery, whereas Fire TV/Samsung/Roku use their own respective stores.

If one supported both the Play Store and their own, the definitions might be more subject to question, but as it stands they don't officially allow that.

It's fairly straightforward, Amazon patches Android, not Linux.

To create their own distinct ecosystem, which is what we're discussing.

Pointing out both Android TV and Amazon Fire is Android OS based isn't particularly relevant to market mineshare. Linux too is used by both (and others), but it would be pretty strange to try to combine them all under a "Linux streaming stick" moniker.

Yes