The most accurate definition of an "indie" game, is one developed by a team of any size, but funded entirely "independently" from any publisher. While a publisher might publish it later, if it's developed and funded by the developer alone, it's an indie game.
The majority of indie games however are from small/one-man teams (in my opinion, quite often likely the final project from game design graduates) and so because these guys code in their bedrooms, funded by themselves or their families we get those old-school, basic graphics "indie" games that I think the OP is thinking of as being the "true indie game". Though in truth, those are also the first thing I think of when someone describes a game as being independent, it's not strictly accurate.
Maybe it's fair to say that while all the basic graphics/old school, simple games are indie, not all indie games are basic/old school. Technically speaking, I would suppose that Star Citizen from RSI and Pillars of Eternity from Obsidian, both Kickstarter funded, are indie, but made with very high budgets by professional teams.
Indie, at best, refers to a company that is not a subsidiary of some larger corporation. Bioware lost indie status when they were purchased by EA for example. The mistake people make is adding assumptions to the mix. Without external finances (from the parent) most indie studios must work with a small staff and limited budget but this is not necessarily true in the long run. Epic was an indie studio by absolute definition and now they are basically quasi-indie given a substantial portion (though not majority the last time I checked) was purchased by a third party company.
Budgets, styles, personalities, etc - none of these universally apply to the group. It is why it makes little sense to try and use it as a classification for anything other than the type of business they operate. When most people say "indie game" they mean "low budget".
It's indie, when the game designers themselves make the decisions.
When there's a publisher involved, or even just a boss in the same company, but not on the development team, then the game and the company are nolonger 'indie' to me.
I see "indie" used fairly informally with a lot of personal definitions in the context of game development/publishing.
So, working under that assumption, I offer my own:
"Self funded, and self-published."
Not Self-Distributed, since by necessity is almost always handled by any number of retailers (physical and/or services).
The practical identifier I use is that an Indie game retains full internal creative control and ownership from start to finish.
Which means any game created or funded via publicly traded companies cannot be "indie".
Notice how I've said nothing about the quality or purpose of the game itself (indie games aren't automatically better than any other; in fact, there are a lot of garbage indie games out right now).
If anything, this post has more firmly shown me that there is no true concensus amongst gamers on what Indie means. I read the Wikipedia definition before I started this thread, and yes I saw where it said:
Independent video game development is the process of creating video games without the financial support of a video game publisher (Citation Needed BTW)
My question though was if there was really more to it than that? It would seem in the public eye there is. When you say Indie game, people tend to have a very real concept in their minds. It may be fair to say that Indie is a broader term that can apply to way more than these Garage Band Developers who make simple or dated graphic games, but because the term is so well associated with those type of developers, I think it's fair you need to have some sort of designation to separate them.
The big reason I assumed there was more to it is because terms like AAA Games which really only means "Popular Game" is used often as a counter-term to Indie, like you can't be Indie and AAA. Whatever the case may be on the true definition, there's certainly a disconnect between what the word actually means and how it's used, which either means the word has been redefined or we are sorely in need of a better word to describe this particular type of game development.