"Since the problem disappears when switching themes, something might be off with your theme indeed."
It worked with Sliding Door at the previous installation.
However, I thought of a thing to try this morning and it worked:
- Copied to a separate file my Custom CSS (because that was a lot of work!)
- Assigned the site to the theme Twenty-Twelve.
- DELETED (not deactivated, but deleted) the Sliding Doors theme.
- Reinstalled Sliding Doors.
- Activated Sliding Doors.
- Pasted my Custom CSS back.
- Success!
So, for whatever reason, some interconnection between the theme and the plug-in got busted during the server move. Deleting and reinstalling the theme solved the problem immediately!
Thank you for helping me stumble in the right direction -- I would not have been able to do it without your good questions!