I pretty much count all games under 20 bucks as indie (if they're new) I know that's wrong, but I don't differentiate beyond that.
Had my most fun last year with those games actually. Mark of the ninja is THE BEST. FTL is also THE BEST.
I do this also which is why I was a bit confused about how to make the distinction between indie and AAA. If we are going with the "smaller game > AAA" option, then they easily win. Some examples:
Super Meat Boy - best platformer this gen, mechanically speaking
Mark of the Ninja - possibly the best traditional ninja game ever made
FTL - the first game to
really make me feel like I was the commander of a space quest
Amnesia - savior of the horror genre. once it hit and was popular, it was like a mini renaissance.
Chivalry - the developer support shames 99% of AAA studios. I feel sick buying map packs now.
Hotline Miami - outclasses nearly every other shooter in terms of heart in mouth intensity
Hawken - the resurrector of good mech combat.
Bastion - a fairly innovative (?) use of narrated storytelling in games. helps that the core game is actually good on top of that.
Trash Panic - Tetris for the 21st century as far as i'm concerned
PixelJunk - too much bite sized casual variety to name
Costume Quest - the most magical game to play on Halloween
Jamestown - so good it got me into SHMUPs, then I found that it's still the only one I like
Torchlight &
Path of Exile - shit on Diablo 3
Natural Selection 2 - arguably some of the best multiplayer gaming ever
Legend of Grimrock - got me into old school dungeon crawling games. now willing to try Etrian games.
World of Goo - a very inventive and intriguing use of physics with a ton of personality
Minecraft - the granddaddy of indie games. digitized Lego-esque crack that even the AAA boys are now aping.
The Walking Dead - I guess this counts. set a new benchmark for engaging characters. one of the best games of 2012.
ARMA - also technically indie. extremely in-depth simulation beyond anything a AAA game could hope for (since it also includes a lot of jank by proxy, something not allowed in 'streamlined' AAA releases). also gave us the DayZ sensation.
Project CARS - another high profile indie title (which will need a publisher on consoles, but only by necessity on there) which has produced some staggering results already using the crowdfunding model. Already murders GT and Forza in cold blood.
... and a cavalcade of other titles like Kickstarter games, AaAaaAA!!, Trials and Joe Danger, Snapshot, Limbo, Capybara Games, Machinarium, echochrome, Shatter, thatgamecompany's games, The Unfinished Swan, Trine, Closure, Braid, Double Fine's games, Jonathan Mak games, bit.trip, Flight Control, Angry Birds, the Playstation CAMP initiative and the pandora's box that is iOS/Android. Aaaand i'll stop, as i'm now doing precisely what the OP is describing.
Basically, what I associate with a small/indie game is often something really unique to that game, like for example FTL is "that perfect Star Trek game", Costume Quest is "that game I play on Halloween", Flight Control is "the air traffic control game", or Hotline Miami is "the awesome unofficial Drive game". You don't often get to make little distinctions and associations like that in the countless AAA games, which are essentially graphics tech demos with some cover shooting thrown in, QTE set pieces, maybe a lite crafting system, and some play-it-safe story fleshed out with ad-hoc text logs scattered everywhere. People complain that small games are just proof of concept titles, but so what? I'd rather play a game completely based around a single inventive idea that is just perfectly executed to its maximum potential, than a game which is a jack of all trades and a master of zilch.