Proton and the Steam Runtime bundle a bunch of different libraries so they all play nice together and are consistent across everybody's machines. There are the obvious things like DXVK and VKD3D, but the Steam Runtime includes basically all of the system files that affect games. It's not quite the same thing but for the sake of simplicity think of it like running in a virtual machine. The Steam Runtime is using libraries from Debian. It is the same concept as docker if you know how that works.

Lutris on the other hand lets you select DXVK and VKD3D versions independently of the wine version, and uses your system's actual libraries rather than the standardized ones. If you're wondering why running Proton inside of Lutris is not working it's because Lutris is missing the Steam Runtime. It's searching for a container that doesn't exist so it can't even start in the first place.