Update
I was able to install a new Joomla 2.5.22 site on the same 1&1 server. Verified that creating a new user and changing the password in the User Profile still allowed the user to login in without CB.
Installed CB 1.9.1 and verified the same thing - all worked.
Matched problem site's modules and plugins with new working site. User can login as long as the password is not changed to the $P$D prefix. Sometimes login doesn't change the password and sometimes it does. If the password is changed in User Profile, the user can't login because of "Incorrect username or password." - the password is $P$D format.
I have reinstalled CB 1.9.1 and nothing helped.
I think the only difference between the two sites, is that I have created registration fields that are not on the working site. I'm afraid to change these on the problem site and lose the info.
Because I'm able to get this to work on 1&1, it is not a server issue. It seems to reside in the changing of the password from User Profile.
Is there a way to do a fresh install of 2.5.22 and CB without losing the registration page, users, and content?
Any idea?