Are you saying that some FTV devices show integer and some show non-integer, during the secret resolution menu cycling? Why would FTV devices act differently? I can understand if it depends on the display device, but sounds like you're saying it depends on the FTV device.
No ... as far as I know the resolution cycling will only cycle through non-integer modes. But as you also mentioned, the screens that you will see during the process will show 24/30/60Hz, although the device is actually running on the non-integer rates. At least that’s how it was on my Fire TV 3 (the pendant). But you can verify this with “System X-Ray” overlay.
And Fire TV devices can of course act differently. They use different chipsets and run different versions of Fire OS. And although Amazon still releases updates even for very old devices, they never up level the OS version on a device. So for example the second generation Fire TV Stick is still running Fire OS 5. And it lacks the “match frame rate” feature, as it’s only available on devices running Fire OS 6 and up. And I also believe that only devices with Fire OS 6+ support both, integer and non-integer refresh rates. At least the Fire TV 3 (pendant) did.
But these are things every owner of a Fire TV device has to check with the developer tools, as there might be differences between other devices as well.
If you want an overview about all Fire TV devices out there, this site will be helpful:
Fire TV device specifications for all models are provided here. The specifications provide more detail than the product pages on Amazon.com, as they're intended for developers b...
developer.amazon.com