The definition of independent film is extremely hazy. Viewed in a certain way, most films are independent, including big Hollywood productions. The studio system, as it was to begin with, is long gone. Back in the day, everyone who worked in Hollywood cinema, from the PAs to the directors to even the stars were merely employees. They were on the payroll of one studio, and they only worked for that one studio. They were assigned to projects, whether they liked them or not.
On an aside, it was the environment of the studio system that spawned auteur theory. Although they were often assigned films to direct, certain filmmakers were able to put their stamp on their work. Directors like Howard Hawks and John Ford made the material they were handed their own, despite their "movie factory" context.
However, the studio system is dead. Now we have the (I believe ill-named) "star system." While it is true that celebrities have an inordinate amount of power, modern Hollywood is based on the freelance philosophy. Actors, directors, writers, and just about everyone else, can work for any studio at anytime. They are hired project by project, and sometimes shot by shot.
The end result is that the big studios merely provide the money, sometimes a place to shoot, and distribution. In some cases one studio may fill one of these roles, while competitors fill others.
The difference between so-called "indie" films and big Hollywood productions? Scale. Nothing more.
One idea of note is that whether a film is considered "indie" or not by the public is often determined by genre and/or what theater it plays in. An example: Reservoir Dogs was distributed by Mirimax. It showed in "art houses." It's considered "indie." Scream was distributed by Dimension. It showed in large multiplexes. It's not considered "indie" even though Dimension is a division of Mirimax.
Like I said, it's hazy.