Thanks for the help, riversatile. I'm not trying to diagnose a DB problem. I'm pointing out what I consider to be a serious breach of my site's security because of a default WordPress error message.
As mentioned in my original post, no one, other than those I choose, should be able to see the underlying file structure of my site and its full DB host address.
As an example, with GoDaddy, the shared hosting path for my public root is
/home/content/zz/xxxxxxxx/html/. This was shown twice on the error page.
xxxxxxxx is a unique number tied to my hosting account.
zz is a number that (I assume) represents a grouping of different accounts. No one on the Internet should be able to see anything above
In addition, a typical GoDaddy DB host address is
my_database_name.db.xxxxxxxx.hostedresource.com. A malicious person, maybe someone who simply dislikes something I wrote on my site, might be able to DDOS my database by simply trying to connect to it with the wrong credentials. I'm not sure what GoDaddy or any other host would do if someone made millions of attempts to connect in a short amount of time. They might be able to ignore it, or they might flag my account - I have no idea.
Think of your database host like your credit card number. While it's not critical for you to keep it secret from everyone, you only want those you trust, e.g., business you write a check to. You don't hand it out to strangers on your business card.