I appreciate your suggestions, thank you. However none of that worked... but I have resolved the 'problem', in an unlikely and somewhat confusing way, read on.
If I remove the module from Magisk then all of the above mentioned directories no longer exist. I checked this after clearing Magisk's cache and rebooting so, presumably, am starting from a clean slate.
If I then install LSPosed v1.6.5 from local storage via Magisk and reboot I am prompted to create a shortcut. If I don't create the shortcut LSPosed does not function (although still visible and enabled in Magisk), no persistent notification is created. I have to reboot to reactivate the shortcut prompt and create the shortcut before LSPosed will work again.
So non buono there but...
Remove LSPosed (v1.6.5) again, install LSPosed v1.
5.3 from local storage, reboot and I have my launcher icon back but my modules don't work (understandable), so I open Magisk and am prompted to update LSPosed to v1.6.5 from the repo (i.e. not "local storage") and reboot... voilà! Now v1.6.5 is working with a launcher icon and no prompt for the parasitic shortcut, all modules enabled and working.
I can't logically explain that.