If you like anime, then you're an anime fan, regardless of which anime you've watched.
If you want to be real anime fan, then you have to watch the non-shit anime that gets mentioned the most often.
If you want to be an anime connoisseur, then you have to be knowledgeable about most anime that gets mentioned, have a few shows that you have decided independently are good shows (i.e. shows that you realised were very good without being told that they were good). A dakimakura is recommended, though not essential.
I'm not sure what the next level up is called, although it involves a complete rejection of media besides anime, manga, and the soundtracks to those anime. It means that they reject real girls in favour of 2d waifus. They only eat seaweed and ricepaper, since it is 2d. They buy VR goggles and high end headphones to be more immersed in their anime. They learned japanese to avoid subs and to experience anime the way the directors envisioned it. They have a katana to kill any intruders, and their food gets delivered to them by mail in envelopes. The walls of their room are covered with posters of their favourite anime and waifu. They earn money by selling used women's panties, however their clients don't know that the used panties weren't used by women. You won't find them here, because on the rare occasion when they communicate with other people (during the start of an anime season), they discuss it on 2chan and on a secret IRC. They've been watching anime for so long, that they no longer get boners when they get turned on, instead they get nosebleeds.