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Have you tried Get-BrokerDesktopName, looked at DesktopGroupName and Name, and see if there is name difference? I'm not sure which name Director looks for. This might be the reason, you are seeing the wrong name in Director.

Yep. On Studio / DB site, everything is fine.

If you don't have to many Citrix Policies and WEM condition/rules pointing at the DG. Remove the VDA's from the DG. Create a new DG and assign the VDAs to that DG and assign the policies for the new DG. Should do the trick.