Here is my list of what truly makes "next-generation gaming":
1) A noticable leap in graphics from one console to the next. This is the most obvious answer. I need to "see" the difference between one console to the next (Example: SNES to the N64, PSX to the PS2 and so on). This doesn't necessarily have to be a huge leap in graphics, but obviously it needs to be worth my buck.
2) A willingness to experiment and explore new concepts and ideas. Hey, life is all about choices. When I play video games, I want to try something fun, innovative and thought provoking. This is one of the main reasons why Splinter Cell: Pandora Tomorrow is one of my favorite games this generation. It literally "forced" me to change the way I thought about playing video games, and it encouraged me to ultimately adapt to my enviorments and surroundings. Obviously I'm talking about its stellar multiplayer mode.
I'm sick of cut-and-dry cookie cutter video games. This was the main reason why I stopped playing games in the first place! There was a period not too long ago where almost everything looked exactly the effing same. You had your Call of Duty's, your Final Fantasys... your Maddens that swarmed the market year after year with very little improvements from it successors and vert little incentive to get me to purchase them. I want something fresh. I want something new.
4) New franchises to ride alongside the old ones. I love me some Halo. I love me some Mario. Hell, I love me some Final Fantasy - but I think it's necessary that we progress into new territories and start some new series and/or franchies that aren't this generations console main-stays. I admit, I never once had the opportunity to try Pikmin.
5) The ability to listen to your customers. Look, we are the people who drive this geeky community. We say what's good and what isn't. We let the developers know in explicit detail what we want out of our games... pretty graphics, refined gameplay and interesting story lines. You can't just go "Well we made this console. We say what flys and what doesn't. You will buy what we put out whether you like it or not." That's not how we work. We need an incentive to purchase your product, and certain things are needed to get us to make that purchase. You need to LISTEN TO YOUR CUSTOMERS, and obviously: respond accordingly.
These days, I could easily fall into the "casual" gamer category. I'm not looking to own the "OMG GREATEST CONSOLE GRAPHIX EVER DUUUUDE." and I'm not looking to own everything "just to own it", but I'm merely looking to have fun. The definition of a true "next-gen" console to me is something that is new, interesting and above all: FUN.