There was an interesting write up over on ComicBookResources' Robot 6 regarding writer Mark Andrew Smith's thoughts on the "indie" tag for creative projects (specifically comics). You can head to the site to see the whole message, but his basic point is that "indie" is a denigrating term that, it seems, equates to amateurism. Smith argues instead for "creator owned" or "creator driven." Robot 6's JK Parkin makes a valid point that those two terms, while possibly able to replace "indie," aren't necessarily interchangeable themselves.
Comic books are rather unique in the sense that, more than other forms of entertainment, the companies to which properties belong really matters. In general it doesn't matter to the public what labels promote what artists or which studio releases a particular movie. For comics, though, Spider-man being a Marvel character is as important as him being Peter Parker. We have the Avengers and the Justice League specifically because these characters being solely to Marvel and DC, respectively. Because of this, and Marvel and DC's complete dominance of the market (more so than other companies are in their industries), independent more or less means "not Marvel or DC." (Smith also addresses this in his post.)
So for comics, "indie" is a term of convenience. What does it mean for other creative projects, though? Is it budget? Mainstream appeal? Star power? It's probably some combination, along with other factors. One thing it is, though, is ambiguous. It's like "underground" or "alt" in that they're hard to define, but you kinda know it when you see it. However, I disagree with Smith that "indie" is a limiting term. In any industry, like with comics, it's used for convenience. Often it does mean creator-owned, especially as it becomes easier for authors, bands, and so on to create and distribute their works without corporate backing and losing their copyright, but not necessarily. It's why the tagline for Upstart Projects - "the best small, independent, and creator-owned projects and creators" - is intentionally broad. I don't want to limit what I should write about because I used the wrong word. Jake Parker's Antler Boy is completely DIY and creator-owned; Mudman by Paul Grist is creator-owned but published through Image Comics, which will automatically give it a perceived leg up in terms of visibility and availability. Is Grist's work less independent because of that? Not to me.
It's an interesting debate, and one not likely to come to a resolution soon. Just looking at the comments on Robot 6, users disagree that "indie" implies inferiority (and note that it's possible to disagree while understanding Smith's stance). In the end, though, it's all just semantics. Creators shouldn't let it limit their creations or creative process. Being categorized doesn't mean being pigeonholed - unless you let it.