When I have issues like this when searching my next search is usually "slack chat" or "zoom video." Now that they're popular, you don't need these qualifiers. What would someone add to "screen"? "screen sharing"?
I don't know if I'm a weirdo but I am so interested in learning about TurboScreenMaster 3000, but it needs to be inherently at least as good as BattleChess 3000 and Class of 3000
All your other suggestions are even more horrific, you might as well call it TurboScreenMaster 3000.
'Screen' and 'Screen Inc' is fine as it is.