Steam has a list of proton versions to choose from in the compatibility settings of the game. When you select one it downloads it automatically. You should only have to download a proton version manually if you want to use the Glorious Eggroll versions, or if you really need a specific version that steam doesn't list, at least as far as i'm aware.

I install the one provided by steam. I guess it should just install the first time you install a game that uses it. And then you will be getting updates via steam too.

I guess, if you are checking out some patches for a game or something like that, then it would be nice to have a simple way to provide your own. But from my experience it's not needed, steam handles proton versions itself

EDIT: proton compatibility layer is something else than just proton?
EDIT2: This is how it looked in the past but AFAIK now you don't even need to enable proton to install proton-only games. Or am I missing something?