Reinstalling the application from scratch did not change the behavior. However, I was able to copy the entire CCS program folder from the fully updated PC to the older one, and the most recent database appears to be in use.

Now, if I select "Live Update" from within an individual remote editor (e.g. MX-850), it says the program is up to date. But if I select "Live Update" from the CCS menu, it says it is still unable to connect to the server. Otherwise, it seems to be working fine. Maybe this installation will encounter a problem as I use it more, but I'll live with it this way for now, as it meets my main objective of getting the latest database onto the older laptop.