Just realizing both versions are in the Play Store (when accessed through a desktop web browser, for the purpose of pushing to a connected Android device - just clarifying that - Chrome does not show up on an Android TV device's Play Store) and both can be pushed out to my Shields and MiBoxes - but I'm not sure which is most appropriate. Any insights?
If you want the Beta, then install the Beta. Or Dev, then install Dev. Or for Canary, install Canary. Otherwise, install the regular release which is titled 'Google Chrome: Fast & Secure'. The extra words in the title are just SEO/marketing speak.
You'll just install it via the web version from a PC.
But Chrome does not show up in the tiles since it's not optimized for TV's, so you have to install another app like Sideload Launcher or use the mic and say "open Chrome" to open the browser.
Chrome beta is as the name says, a less stable more updated version of Chrome. There is also Chrome Canary
I use Firefox for Android TV. Fast and simple.
One pro tip for you guys. Actually you don't even have to use your pc to push the chrome.
Just say "open chrome" in the mic and it will take you to the 'hidden' chrome page in AndroidTV play store app and you will be able to install it from there.
I am not sure about other boxes but it worked on MiBox (stable firmware).
You can use TV Bro web browser:
https://play.google.com/store/apps/details?id=com.phlox.tvwebbrowser
I use Fast and Secure on my Shield.