It's a game that's developed and published by one studio without the aid of a larger publisher. This is the correct definition. It's literally what "independent" means. We only call smaller projects indie games because we don't have a good word for them.
Kind of like with indie music. It was music that was distributed without a major label, but we didn't have a word for the genre they were commonly doing, so we started associating the sound itself with "indie." I feel it's a bit different with games though because indie music had a particular style, whereas games labeled as indie are really just smaller than regular games, sometimes even made by huge publishing companies.
Most people here want to do the whole "it's different for each person" thing, and I'm normally up for that line of thinking, but in this case there really is a correct definition for the term, and we're just plain using it wrong a lot of the time. (I'll probably get a lot of crap for being so sure of myself, huh?)