It would be really nice if they would update the documentation on this. I've been fiddling with JWT tokens for 5 hours. Shouldn't it return 4040010 instead of 401 in this case?

It might not have been before but it is now. When you update the Bundle display name (CFBundleDisplayName) it should be changed. It looks like the display name is not updated if you add a widget to your homescreen and then change the display name, but if you delete the widget and add it again the name should be updated to match the bundle display name of the main targer.