Talking to a teenage Colombiana last week, I used the term, la red, for the Internet. She corrected me and said that the term is not used anymore. Even in español, one uses the word Internet. There are so many regional differences in this language, however, that you better check with a local to see what the practice is in his or her location.
I also encountered hairnet as "la red" the first time. The slightly absurd association made me remember it very well. I think this is a practice DL should utilise more often because it is a very effective way to learn a word that has multiple meanings; the more bizarre, the more it sticks, and the easier you'll recall a semantic understanding of it.