I think the biggest problem is the definition of "generation" is applied too freely. How is the PS2 a "generation" ahead of the PS1? What does it offer in terms of new experiences over sustaining innovations? It's basically a souped-up PS1, for all intents and purposes. How is the PS1 a "generation" ahead of the SNES? The jump to 3D, perhaps? No, 3D is just another form of graphical improvement; it doesn't change the experience, it just expands the existing experience. Is the SNES a "generation" ahead of the NES? Not hardly; the SNES is as much a souped-up NES as the PS2 is a souped-up PS1. The common thread between these "generations" is only improved hardware capabilities. But as you've already noted, hardware capabilities don't define a generation! So in fact, the PS2 is still in the same real generation as the NES.

So what does that leave to define a generation? Well, let's look at the Atari and NES. What sets those apart? Ah, now there we can find something concrete that doesn't involve improving technical specs! The Atari's controls and games are of a completely different mindset from the NES! Indeed, Atari controllers are held in a totally different way, and the games are all very different: simple and largely static. They'd practically be termed "mini-games" by modern standards, and usually not very good ones, either. So perhaps we have something there to mark as a generation: a change in how games are played and what type of games are played.

By that standard, let's look at things as they stand now. The PS3 and 360... pretty much go by the same model as the SNES, PS1, and PS2 did, with the usual sustaining upgrades. They're still using the same controls, and still providing the same types of games as the NES onwards (with the same types of upgrades along the way). The Wii, however, changes the controls and the types of games being played. Which fits the definition of "generation" above perfectly! Imagine that.

Not true.

As noted above the least powerful system has not always won.

NES was > all 8 bit systems.
SNES was > Genesis while being PS1 was > Saturn while being PS2 was > Dreamcast while being Wii
All this proves is that the actual technology isn't the most important factor. Because the only thing all of these winners have in common is that they all had the greater games library.

Games sell systems, and like all previous gens, in the end the Wii will have a much larger library of all sorts of games.

I suppose that still leaves the question of how you can determine who "wins" a particular round of systems within a generation, huh? Well again, technical specs don't matter.

Marketing, perhaps? Hm, Genesis was marketed as heavily as (if not more heavily than) the SNES, and MS, Sony, and Nintendo all were quite heavy on marketing last generation, so... no.

First or second out the door? Let's see, SNES came out after Genesis, PS1 came out before all competitors that iteration, and PS2 came out after DreamCast. So no, that doesn't matter either.

How about initial game library? The SNES' library out the door was sizeable, but no match for the existing Genesis library at the time. And of course DreamCast had a much bigger library in place already when the PS2 launched. So it's not the launch library.

Lifetime game library? Well, yes, the lifetime libraries of systems which won their iterations were impressive, but much of that support came after the fact. So this isn't a fair criteria to judge by; developers will develop for the system they feel is going to "win", and there has to be a "winner" in their minds before that can happen.

System price? Well, the PS1 cost more than the N64, but there were a lot more PS1s sold. And the GameCube was cheaper than the PS2 for most of last generation, but PS2 still dominated. So no.

Brand loyalty? Ahahaha, no. If that were the determining factor, then Nintendo would never have lost support after the SNES.

Adoption rate? Well, there's definitely something there, but it's only part of the answer. The SNES, PS1, and PS2 all experienced faster adoption rates than their competitors. But why? Let's keep delving deeper...

Distribution base size? Ah, I think we may be onto something there! The SNES and Genesis were sold to largely the same markets, and ultimately ended up with close sales; where Nintendo dominated, Sega faltered, and vice versa. Ultimately, Nintendo's triumph over Sega was one of having a larger distribution base sooner in Japan and America. Sony released the PS1 in more countries than Nintendo did the N64. The PS2 similarly was launched in many territories as fast as possible. This model fits each iteration, and gives a definite answer: what determines if a system "wins" an iteration? Distribution base size. There could be more than that, but if there is, it's not part of the above.

Factors:

B-titles: These games come out enmasse and meet the many different tastes of the market.

Shovelware: These games are usually geared towards kids and non-savvy gamers who want a specific action or gimmick for 20-30 bucks.

Casual titles: Games that the motivating family member can play with other family members (NES - Duck Hunt, Track and Field, etc.)

Value: NES had a great package, SNES had the name nintendo and a good package, PS1 was a game player and a cd player, PS2 had the name, was a gameplayer and a DVD player.

All of them (except SNES) had a pretty good selection of cheap games by the third year of the cycle.

People, unfortunately, want quantity not quality. Especially in America with the Wal Mart Shopping and the all you can eat buffets. (I live in the US)

One thing you have to keep in mind is that, for a success model to work, it has to apply properly to all examples: past, present, and future ones. If your model for success fails to apply for even one example, then it is not accurate.

cheaper so that they can sell each console and make a profit unlike the PS3 and Xbox 360(actually the xbox 360 is starting to make profit on each console sold now)

I think you don't understand the term generation. Are you a generation ahead of your parents? What software and hardware innovations did you bring? Probably not many.

You are putting the cart before the horse. Success brings the quantity of titles, while failure results in eventual loss of titles (as we saw at the end of the N64 and GCN's life cycle).

And as the PS2, PS1 and SNES showed, quantity will bring quality, and most likely hit a wider variety of consumer interests in the process. 

Then perhaps "generation" is not the correct term at all. Perhaps there is something "above" generation, something that fits what I laid out there. An era, perhaps? That seems like a fitting enough term for it: eras often span many generations, and are marked at beginning and end by significant events that end the previous era and introduce the next.

It seems you have forgotten the nes was about 5 years behind it's competitors graphically.