Our script just determines LS_HOME by looking at the path of the launched script itself and moving up two directories.
So it seems that you have run something like:
/Lightstreamer/bin/unix-like/background_start.sh
which is not what we expect from the launch through init.d.

After running the install script (the "Debian" version for Ubuntu), you should have a properly configured /etc/init.d/lightstreamer script file.
May you please attach this file (perhaps in txt form) for a check?