You can have Bridge as an additional organizer for Photoshop during working with Camera Raw on Photoshop.
Whereas Lightroom software works as an image processor that uses a lot of tools for manipulating the image in it.
Does that mean Lightroom Classic is in fact the cloud version and I am mistaken? I'm not certain how I missed out getting version 7, I have no memory of there being a release.
If you are running Lightroom 6.14 (not the "last version of Lightroom Classic", which is currently at version 9.1), I assume you are using an older version of Photoshop. The message means that because Lightroom is at the 10.1 version of Camera Raw, when you choose to edit in Photoshop (which has its own Camera Raw plug-in), then Photoshop's Camera Raw plug-in should also be at version 10.1 or later, and that message indicates that it isn't.