I solved my problem. Here's the story:
I searched for "LU1863 disk space" in Google. I came across a Russian language post. Out of desperation I read down through the postings after that entry. I saw the words "ISA Server 2004" and "GFI WebMonitor". Ahah! I also am running those products. My version of GFI WebMonitor was not current, so I thought it could be causing the problem. I uninstalled GFI WebMonitor because I no longer need it. No restart was required. Now LiveUpdate works as it should. It no longer generates the LU1863 error.
My conclusion is that anyone getting a misleading LU1863 error should very carefully consider all the products they have installed that filter or block web content. GFI WebMonitor inserts an extra step in the download process so it can virus scan the downloaded file before releasing it to the user. I assume this extra step causes the LU1863 error because it blocks direct downloading (unless the site is whitelisted within WebMonitor).
Thank you to everyone who has analyzed or assisted me with this problem.
Good luck finding the blocking program that's causing your LU1863 error. Please be sure to post your solution here.
Thanks,
Sherman Colson