I think "indie game" is a useful distinction for determining where a game is available and what level of access you may have to the developer (how likely they are to respond to feedback directly and whatnot). Are there exceptions to the rule? Absolutely, but it's still a generally useful metric.
Distribution: I wouldn't classify Valve as "indie" any longer because they own the means of their distribution. You'll never find Team Fortress 2 on Origin or GOG. I'd give the same argument for thatgamecompany: at the time Journey was developed, thatgamecompany would not have qualified as an indie developer because they were tied to the money and platform Sony provided.
The line is fuzzy, but indie games tend to be found in freer markets. Steam's Greenlight program, Microsoft's "Indie" section, and if you don't like a given distribution method or platform, you're likely to find it elsewhere. Thomas Was Alone is available on just about everything, and Mike Bithell has final say over where else it gets ported as the developer. Don't want to buy it on Steam? Get it from his site, from the Humble Bundle, from Desura.
Access: Indie developers tend to be much more available to the general public. They'll have discussions with their fans, they'll incorporate feedback more readily, they'll get in Twitter arguments and join causes. AAA developers tend to be more locked down, controlled by PR. I had a podcast with 100 subscribers, but I could probably get Zoe Quinn on to talk about Depression Quest. That's not going to happen with CliffyB.
I think that may also be why indie games have a bad rap. They're more likely to take on issues and challenge convention than a game with millions of dollars behind it. Moreover, the developers are more likely to voice and stand behind their reasoning, rather than try to claim they're not making any "social commentary." It gets them in trouble because the general public tends to dislike the status quo being challenged, no matter how much it needs to be.
Which, I think, is why looking at how a term is perceived is a bad idea. Game/Show has done this twice now (that I've seen) with "gamer" and "indie games." The argument seems to be that the public has a negative perception of the word, so abandon the word. That's easy, but counterproductive. The public had a negative perception of rap and metal music for a while, and rappers sometimes sing and metal bands sometimes have short hair, so let's just abandon the word and call it all "music."
It's not a perfect analogy, but the point is that you don't just abandon a useful phrase because it's poorly perceived. If people are using the term "indie game" then it maintains some utility. So, it falls to us to better define the phrase, and to remember that exceptions often help define the rule, rather than denying it.
I think "indie game" falls on the class, or maybe even phylum, level of classification. If so, the number of assumptions you can make about an indie game are going to be relatively few. If we start using the phrase appropriately, perception will change, as it always does.