I solved this. I looked in /var/log/apt/history.log to see what had been recently uninstalled by searching for the "Remove: " lines.

I found that all these packages had been recently "helpfully" selected for automatic uninstallation, and they had not all been installed.

I manually reconstructed a package list of the removed packages, and passed that  "apt-get install". Sure enough, several more packages were found to be re-installed. 

Then, after rebooting, my video drivers were working as intended. 

Here's what I ran:

sudo apt-get install xserver-xorg-video-mach64-lts-quantal xserver-xorg-video-trident-lts-quantal xserver-xorg-video-sis-lts-quantal x11-xserver-utils-lts-quantal xserver-common-lts-quantal libgbm1-lts-quantal xserver-xorg-video-modesetting-lts-quantal xserver-xorg-input-synaptics-lts-quantal xserver-xorg-input-evdev-lts-quantal xserver-xorg-video-ati-lts-quantal libxatracker1-lts-quantal libgl1-mesa-glx-lts-quantal libgl1-mesa-glx-lts-quantal libgles1-mesa-lts-quantal xserver-xorg-input-vmmouse-lts-quantal xserver-xorg-video-s3-lts-quantal libegl1-mesa-drivers-lts-quantal xserver-xorg-video-vesa-lts-quantal libopenvg1-mesa-lts-quantal skype xserver-xorg-video-tdfx-lts-quantal libegl1-mesa-lts-quantal xserver-xorg-video-fbdev-lts-quantal xserver-xorg-video-nouveau-lts-quantal libglapi-mesa-lts-quantal libglapi-mesa-lts-quantal xserver-xorg-video-mga-lts-quantal xserver-xorg-video-sisusb-lts-quantal xserver-xorg-video-vmware-lts-quantal xserver-xorg-video-neomagic-lts-quantal xserver-xorg-video-radeon-lts-quantal xserver-xorg-video-openchrome-lts-quantal skype-bin xserver-xorg-video-cirrus-lts-quantal xserver-xorg-input-wacom-lts-quantal xorg xserver-xorg-input-mouse-lts-quantal xserver-xorg-input-all-lts-quantal xserver-xorg-video-intel-lts-quantal xserver-xorg-video-savage-lts-quantal xserver-xorg-video-all-lts-quantal libgl1-mesa-dri-lts-quantal libgl1-mesa-dri-lts-quantal xserver-xorg-video-siliconmotion-lts-quantal xserver-xorg-lts-quantal xserver-xorg-core-lts-quantal libgles2-mesa-lts-quantal xserver-xorg-video-r128-lts-quantal