You appear to have stumbled across a slightly different website for the different campsites in Tenerife to the one I'm referencing. Both have the same appearence, but yours is www.cabtfe.es, whereas mine is tenerife.es .
The one I am using contains a link to obtain the camping permit, so I think it's slightly more official than the one you looked at.
There are, rest assured, 20 campsites on the island.
Las Lajas has a quite high altitude, so, in winter, you should certainly take extra blankets to keep you warm through the night.