Besides, I can easily see every body in the solar system fitting into one of three categories: rocky, gassy or icy.
Not really. For one thing, the concept of a "gas giant" is somewhat outdated, since we now know they're mostly liquid. "Giant planet" is the preferred formal term. Also, Uranus and Neptune are significantly different from Jupiter and Saturn, a subclass sometimes called ice giants because their chemical composition includes water and other compounds that are normally ices in most of the outer system. In fact, if Uranus and Neptune had migrated closer to the Sun, they could've lost their atmospheres and become members of the class known as ocean planets, consisting maybe 50% or more by mass of water/ice over a rocky/metallic core. Essentially they are icy/watery worlds, just ones that have thick hydrogen atmospheres too.
Yes. As I recall, there were several drafts of this IAU proposal up for a vote, and there was an "a" and a "b" version of the dwarf planet proposal--the "a" said that dwarf planet was a sub-category of planet, and the "b" version had dwarf planet as its own category that were not real planets at all; and the IAU voted almost unanimously for "b."
Actually I think what happened is that there was an impasse between "a" and "b" so the IAU subcommittee slapped together a "c" at the last minute, and that was what was actually presented for the members as a whole to vote on. The "c" proposal was an attempt to compromise between "a" and "b" and address other concerns, but it was what you'd expect from a rushed compromise solution arrived at by a committee.
That is the kind of thing I was referring to--that the goal of this reclassification (at least amongst the people voting) was to find any excuse to remove Pluto and all the other small bodies from planetary status, whether it made any sense or not.
It was about deciding whether or not it was acceptable to let the number of planets in the system swell into the dozens, even hundreds, or whether it was better to keep it to a limited number so as to preserve tradition or avoid "devaluing" the concept of a planet. The latter way won out, and I think that was choosing sentiment over science. The vast majority of stars in the galaxy are tiny, dim red dwarfs, but that doesn't devalue the Sun or Rigel or Antares.
Assuming this isn't sarcasm (

), I agree. I recall as a child when I was first learning about the layout of the solar system that it seemed very odd to have these couple of spherical "asteroids"--Ceres and Eris--out there, but not calling them planets just because they happened to share the neighborhood with a bunch of irregular rocks.
Err, Eris has only been known to exist since 2005. And Ceres is the only really spheroidal asteroid in the Main Belt, although Vesta probably would be if not for the huge impact crater that gouged out its south pole and left it shaped more like a flattened pear or something.
And what about the
Trojan and Greek asteroids that share its orbit? The IAU ruling is trying to exclude dwarf planets because they don't have enough gravity to pull in all the nearby rocks, but what about planet that has so much it concentrates debris along its orbit at Lagrange points? Or do those not count just "'cause"?
The proponents of the "cleared orbit" definition point out that the Trojans are still controlled by Jupiter's gravity, so that Jupiter is the gravitationally dominant body throughout its whole orbit. But that is a bit of an awkward way of expressing it, and as stated, it's too orbit-specific; take the same body into a further orbit and it wouldn't be able to wield the same gravitational dominance over the whole thing (simply, I would imagine, because the orbit would be so much bigger).
I don't know-it's been a while since I saw that segment, but I got a much more hostile impression off of him than that. As David pointed out, Dr. Tyson had Pluto removed entirely from the planetarium display years before the IAU vote, which matches up with my impression of him as, essentially, holding a grudge against Pluto. Which is a really, really unscientific thing to hold a grudge about, but there you go.
I don't think it's a grudge, it's raising a valid scientific question. Pluto has always, always been unlike the other planets. Its orbit is anomalous, it's a small rocky/icy body out where we'd previously only found giant planets and moons, etc. Astronomers only called it a planet in the first place because they didn't know what else they could call it, but from the beginning its exact nature and identity were a matter of uncertainty. Once we began to discover other Kuiper Belt bodies, starting about 15 years ago, we began to see a possible explanation for Pluto's anomalies, and many astronomers began thinking of it as sort of a large cometary body rather than a planet. The IAU thing in 2006 wasn't out of the blue, it was a culmination (though clearly not a resolution) of years of growing debate within the astronomical community.
So it's got nothing to do with grudges. It's silly to say that just because something isn't a planet, that means it isn't as worthwhile. The cosmos is full of fascinating things that aren't planets. The idea that there's a whole new belt of cometary bodies beyond Neptune is a wondrous, amazing notion. It's not about putting Pluto down, it's about caring enough about Pluto to want to understand what it truly is rather than just shoving it into an old category that may not do it justice.