Weird SEO Question...

First are you sure you are seeing the true google results and not your personalized ones? Remember that google personalizes so your version of results is not always the default. This is the reason people seem to find their websites on page one but no one else does.

Second I wonder if site one has lots of "no value" links. Remember google is smart enough to know people game the system. So the middle site might have 178 true and natural links while site one has 3k plus "fake" links. Hell google may have even punished them if you truly don't find them at all.

SEO is a lot more than links. In many ways the best SEO is similar to good PR. You get the right people talking about you and people (and google) notice. But just like PR you can do a lot of things to get your name out that no one notices. Such as Press Releases that are not really news but advertising (some news sites will auto load these but they will never be turned into an actual news story). A lot of old school SEO seemed to follow that kind of idea. That getting your link out there was all that mattered. But Google is smarter than that.

Remember google needs to provide the best content to their customers or people will stop using them. So they build systems to find the best content and every month those systems get better. Can they be gamed? Of course but over time it becomes harder and harder to game them.

Seems like it would just be easier to find ways to provide great content that gets shared naturally so google finds it, doesn't it? Almost like SEO really is about the end user (the people searching) and not about getting to the top of google. But you know me I say crazy stuff like that.

I mean it must be smarter to just game the system, right? Why would people keep doing it and getting slapped back several pages in rank when the systems change in major ways if it didn't make sense? /sarcasm

What is the term you have made these comparisons for? You simply may need to look at a wider range and the first site may actually get a lot more exposure in search results as for other terms.

Most of that is meaningless.

Compare the on page optimization of the keyword. Of the backlinks, how many have the anchor text with the targeted keyword?

Consider the ratios.. the site in the middle actually looks the strongest.

Site 1: 3,671 external links... only 364 root domains. Basically 10:1 for unique domain backlinks.
Site 2: 178 externals with 58 unique roots. 3:1 ratio
Site 3: 6.5:1 ratio

What the chart shows me, is that site 1 has a lot of spammy links. Site # 3 has a lot of careful but misleading linkbuilding, with the unlikely amount of followed links and no nofollow. Site #2, has the best ratio of unique backlinks and also a very natural link profile from what you show.

All in all... don't take what seomoz shows to heart... their authority and trust rank algorithm is much different than Google's. Also, make sure to compare the on page optimization, as that usually shows the full story, but from you show the link profile of site 1 looks like someone that has ordered waay too many gigs on fiverr. lol.