If you look up the roots of the word fan it is very instructive at least for what the word used to mean. According to common dictionaries "fan" is the shortened form fanatic. Fanatic came into the English language around 1550, meaning "marked by excessive enthusiasm and often intense uncritical devotion". The root of fanatic comes from the Latin fanaticus, meaning "insanely but divinely inspired." Especially note the "uncritical" part of the definition.

Unfortunately, perhaps due to the internet, the word fan seems to less and less mean an aficionado, supporter or person who is enthusiastically devoted to the team. More and more the word fan seems to come from the word fantasy. People are imagining they are coaches pronouncing what players should be on the field and what plays and formations should be run. If that wasn't enough, now they fancy themselves as administrators worthy of deciding when and whom should be fired and how. I guess you could include imagining themselves as sports writers as part of the fantasy world they live in. Perhaps even their purchase of a few tickets and related sports memorabilia a few times a year qualifies them as a big booster whose opinion should be counted. I prefer to be what a fan used to be. Supporting our team and coaches because it is our team and our coaches not because they are winners or losers.