Open Preferences and type icon options inside the search box. Set the settings that appear to your liking. If they are already set as you prefer, toggle them to their opposite state then back to the desired state and see if it restores the right behaviour. Additional options are available when you click the App Icon Options button.

I have tried toggling the icon options, restoring defaults, even rebooting. But it still seems to go back to this behavior. I do get an audible warning when new mail arrives. There are a large number of filters on the inbox, which move the mail into various folders, so almost nothing stays in the inbox. I do get the icon (with a red dot) in the notification tray and a count in the taskbar icon, only if it stays in my inbox. I really want a new mail notification if the mail is not marked as read by the filters, in any folder (as before).

I think the issue is that it is new mail in the inbox that both things look at.

Or that the developer was in the wrong frame of mind when they decided that people only use the inbox to check mail. I mean, can we get an adult to fix this??

I mean can we get an adult to fix this??

That's uncalled for.

But then so is your comment. Redundant.