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.