Some progress has been made.
Today, I have gone to the TTY1 console, logged in to my account, stopped the
LightDM service, and started X. I looked at glxinfo, and saw the correct info
instead of llvmpipe. It now says that I have "DRI Radeon R200..." in renderer
string, and when I started Compiz, it is smooth.
I even added lightdm user to the video group because I suspect that lightdm
is starting X server with the session of lightdm user, but it didn't change from
llvmpipe to DRI Radeon.
I don't want to kill lightdm and restart X from TTY1 to have a usable accelerated
desktop, so I was wondering if there is some kind of configuration in lightdm
that would enable hardware acceleration. If not, then what should I do to
force lightdm to use hardware acceleration?