The period for support doesn't depend on what was initially installed but solely on the package.
A package that is considered to be a "server package" gets LTS on any installation. Packages that are considered both "server packages" and "desktop packages" get LTS as well. Packages that are considered "desktop-only" don't get LTS on any installation.
So during the server LTS period on a desktop installation you still get updates for some packages, but not for others.
For example it's still Firefox 20 in Ubuntu 10.04 with all its known security bugs.
Please upgrade to a still supported version of Ubuntu as soon as possible.