Amazon Prime App on Apple TV

Amazon Prime Video is available for:

Apple TV (3rd generation) with Apple TV Software 7.2.2. Amazon Prime Video is automatically added there.

Apple TV (4th and 5th generation). Amazon Prime Video is downloadable in the tvOS App Store.

Alternatively, one could AirPlay from the similar app on an iOS device to Apple TV as well.

That makes perfect sense. 🙂

I have feeling I have an earlier generation of Apple TV than the ones mentioned.
Purchased in 2008. Version 6.2

The 1st generation (A1218) was on sale from 2007–2010. Its final software version is 3.0.2.

The 2nd generation (A1378) was on sale from 2010–2012. Its final software version is 6.2.1.

The 3rd generation (A1427, A1469) was on sale from 2012–2016. Its final software version is 7.2.2.

All these tidbits seem different from your statement?

How does one get to the tvOS App Store from the TV?

The App Store is only available on the 4th and 5th generation, with software version 9.0 or later.