> Cortex-${Foo} as a brand-name for a particular type of CPU core.

You could be more specific and say it's the brand name that Arm holdings uses for their CPU designs.
This is less of an issue for R and M cores, where there aren't any Arm architecture CPUs designed by other companies (that I know of)

> The Cortex-A53 meanwhile implements the Armv8-A Architecture and supports both Aarch64 modes and Aarch32 modes

and there's other Armv8.x-A cores that only support AArch32, and others that only support AArch64
Armv9-A mandates cores that *must* support AArch64 (at least at certain exception levels) - sounds like fun.

further reading, in case you enjoy these sorts of things:
https://maskray.me/blog/2023-03-05-linker-notes-on-aarch64
https://nickdesaulniers.github.io/blog/2023/03/10/disambiguating-arm