problems with touchscreen plugin and hdmi connected display

I want to connect my raspberry 2 model B with a monitor (later it will be a touchscreen).
According to the instruction that can be found in … Guide.html
I installed the touchscreen plugin and activated it. After that I did a reboot, but on the monitor connected to the raspberry I still get only the login prompt, no X-server or similar has been started.

What I have seen during the installation of the plugin, was that some of the actions could not be finished, the installation procedure was complaining about not having enough space on the microSD card. But it reported that the plugin was installed successfully.

The sd card is a 16GB card, and there is a lot of space not used.

When I checked the used storage spaces with the “df” command, I found that the “overlay” filesystem with mountpoint “/” has only 2% space left, but I don’t know how to figure out to which partition this mountpoint is connected to, to provide more space to this partition (if this is the root cause of my problem).