It is not possible to do a desktop files rebuild for the "Unix" volume (the shared folder), so rebuilding the desktop will not correct icons of applications that reside in the "Unix" volume (the shared folder). Note that also files dragged from the "Unix" volume onto the desktop are still on that volume
That explains it perfectly! Excellent answer - I understand, and of course your advice worked. To get the icons, I copied from the shared volume onto the real volume (the startup volume) and did a desktop rebuild.
Amusingly, as a side benefit, this also
caused the copy of the app and its documents on the Unix shared volume to achieve the correct icons.
Running the app from from the startup volume also solved a mysterious file system error I had been getting, so I'll work that way from now on. Thanks for clarifying this!