First of all, I'm not a movie expert. By this I mean I don't go to film school. If film school is anything like business school, you don't actually have to go to be a successful businessman, although it would be tough to be an expert in securities valuation without taking Investments. In any case, my only credentials are that I read reviews fanatically and watch every movie I can in the 24 hours in a day. Roger Ebert is my idol and in my opinion a revolutionary in the field of criticism. I thought this was a necessary disclaimer for anyone who likes to "check their sources."  Take my opinion then, if you choose, with a grain of salt. Here we go.
I think it's ironic that the largest distributors of independent films are subdivisions of large companies, take Fox Searchlight for example. As a business person I think this is great, you get independently made films being distributed by companies with a ton of resources and power which translates into more widespread distribution. I will admit to not being the biggest follower of independent films, but only because I think a movie benefits from an experienced director. I'll take one example that underscores my point there, and then present a contradiction.
Paranormal Activity is one of the worst movies I remember seeing in a long time, that was mostly praised by critics. I've never been a fan of horror films in general. People say they play on our fears, but how can I be afraid of something that is so moronic and unrealistic. Movies should inspire real emotion. Paranormal Activity gave us a movie that had no plot, acting, characters, or realism. It was a series of stunts performed cheaply that were designed to shock the audience. I'm feeling a reference to a Victorian Freakshow is in order. But what I'm trying to say is that Paranormal Activity was just a show, and not a film.
On the other hand, look at Clerks, Kevin Smith's first film and in my mind a masterpiece of low budget film making. Clerks had a lot of things that Paranormal Activity did not. The most extraordinary thing about it was the dialogue, filled with social and sexual commentary, humorous and yet insightful observations, and overall extremely witty and creative. It showed a different way of life, a different way of thinking about things, exactly what a movie should do. If Kevin Smith's brand of humor may not appeal to everyone, try and see past the jokes and into the writing. The acting was where it should have been and special effects were budget prohibited, but the writing was pure and simple genius.
So what does this say about my point on independent film? I think it emphasizes that it's mostly hit or miss. You may have a rising prodigy (Smith) or someone who knows how to hustle us with the camera (Peli).
Classifying films as independent is also tricky because some of them may have a big name director involved (Steven Speilberg in the case of Paranormal Activity) but still be called independent.. Sure it refers to the funding and studios involved, but if Wal-Mart provided the funding could a film still evoke the same feelings as a really independent one?  A change in the definition of independent needs to happen and it has to do with just that.