I will attempt to answer my own question...
It would see that we need to duplicate existing content, then click on Visibility on the right sidebar. So I would have my original "row" Visibility ENABLED when viewing on in Desktop & Tablet views, but DISABLE it for the MOBILE view. And then I would select my duplicate row and DISABLE Visibility for Desktop & Tablet but ENABLE Visibility for MOBILE. Then I can independently edit content for various views, although the design area gets a little messy when doing that.
I still can't figure out what that rightmost "eyeball" icon is for though. It sits to the right of Desktop, Tablet and Mobile icons in the right sidebar under Visibility. I thought it would make the selected content vanish from the design view (making it cleaner), but that doesn't happen. Is this a bug in 2.4b3?
Speaking of possible bugs, I am unsure why alignment is so inconsistent in design view versus browser preview and why Header-tabbed box content differs in alignment from A-tabbed boxes. This is shown in my screenshot here: