Designed for the fourth and fifth-generation Apple TV models, the new tvOS 12.1 beta can be downloaded onto the Apple TV via a profile that's installed using Xcode. Subsequent betas can be downloaded over-the-air.
We're not yet sure what fixes and changes the tvOS 12.1 update might bring, but it's likely to focus on bugs that have been discovered since the release of tvOS 12. No new changes were discovered in the first two tvOS 12.1 betas.
Apple's tvOS updates have historically been minor in scale, and Apple does not often provide us with detailed notes outlining what's new, but we'll update this post should any changes be found in the third beta.
Update: Apple has also seeded a new beta of tvOS 12.1 to public beta testers.