Part of the problem here is semantics. The meaning of "planet" was really poorly defined prior to 2006 (and some argue that it still is). In the early 2000's, this started to become an issue. You can blame Mike Brown.
In January 2005, Mike Brown's team at CalTech in Pasadena, California, discovered an object we now know as Eris (they nicknamed it Xena early on). The early observations of Eris suggested it was larger than Pluto (which was later conclusively proven incorrect with the New Horizons flyby of Pluto in 2015). But larger or not, it was close in size to Pluto. Did this make it the tenth planet? If so, then what about the other bodies like Sedna? Haumea? MakeMake? For that mater, what about Ceres, or Chiron? By some counts, you could say we had upwards of 23 planets. Or do we only have 9? Or 8?
For that matter, what makes a planet a planet?
The word planet comes to us from the ancient Greek "planetes asteres" which meant "wandering stars." To the ancients, even well before the Greeks, a few of the lights in the night sky were definitely different from the others. The stars remained in fixed patterns. Sure, they rose and set, and they rose and set at different times through the year, but their relative positions were always the same - as far as the ancients could tell. But five of them were different. Their positions changed. Two of them would rise before the sun in the morning for a while, then disappear and set after the sun in the evening for a while. These are Mercury and Venus. The other three, Mars, Jupiter, and Saturn, all seemed to move in the same direction, interestingly through the constellations they knew as the zodiac, but now and then they'd back-track a little bit, then move forward again. Very odd.
The ancients had no idea that they were, in fact, other worlds, completely different in nature from the burning stars. Of course, for that matter, they had no idea what the stars really were either. They just knew the planets, though they looked pretty much the same as stars, moved differently. They were wandering stars. It wasn't until Galileo turned his scope on the heavens that we learned that these aren't just bright points of light like the other stars - but whole different worlds. The ancients barely could see Uranus, and gave it little heed, and couldn't see Neptune at all. Other, smaller bodies like Pluto, Ceres, and others were far to small to see without a telescope.
When Herschel discovered Uranus in 1781, he first thought it was a comet. In 1783, observations and calculations had lead to the discovery that Uranus' orbit was nearly circular, and it behaved like the other planets, and so it was accepted as one (and Herschel wanted to name it George). When the predicted movements of Uranus didn't match the observed motions, astronomers and mathematicians suggested the reason was due to another planet, further out, pulling on Uranus and affecting its orbit. In 1846 it was finally discovered, largely based on mathematical predictions. At the time, no telescope could show either Uranus or Neptune as more than a very small disk. But they were in nearly circular orbits, and that seemed to be good enough.
Ceres was discovered in 1801, and for several years was considered a planet. German Astronomer Johann Elerte Bode had previously observed what he thought was a mathematical relationship between the planetary orbital radii. According to his calculations, there should be a planet between Jupiter and Mars, right where Ceres was discovered by Guiseppe Piazzi. Bode's observation was actually erroneous - there was no such mathematical realationship, but it was convenient that an object was found there.
As it turned out, Neptune's orbit didn't quite fit the mathematical predictions either. This lead to a belief that there might be a tenth planet causing the orbital discrepancies. In the late 1800's and into the 1900's, the hunt was on. Wealthy eccentric and astronomer Percival Lowell founded the Lowell Observatory in Fagstaff, Arizona, and set out to find "Planet X." In 1929, the director of the Lowell Observatory hired a 23 year old farmer from Kansas who'd been constructing his own telescopes, mostly out of salvaged farm equipment, and doing his own observations. Clyde Tombaugh was brought in to work on the hunt for Planet X, a rather arduous task involving taking pictures of the same patch of sky a few days apart and then comparing them using a device called a "blink comparator" and looking or something different between the two images. On February 18, 1930, Tombaugh noticed a tiny dot moving between images taken a few weeks before. Looking back at a previous image a few days before, he added to the evidence. After capturing a few more images still and comparing, he finally reported the discovery on March 13, 1930. He had discovered Pluto - the ninth planet!
Well, at least, that's what the hype said. It didn't take too long to notice that Pluto was a bit different. While no two planets orbit in the same precise plane, the previously-discovered eight are very close to co-planar. They also have orbits that are very close to circular. Pluto's orbit is tilted dramatically with respect to the other planets, and its orbit is significantly more elliptical. In fact - Pluto's orbit crosses inside that of Neptune (though they'll never hit each other).
Pluto was also tiny. Our own moon is larger, as are several other moons of other planets. In 1978, Pluto's moon Charon was discovered. With further research, it was found that Charon was so large, that the point of orbit between them, what is known as the barycenter of the orbit, is actually well outside Pluto's radius. Many astronomers were starting to talk of Pluto-Charon as a "co-planet."
Then, in the late 1990's and into the 2000's, a bunch of other objects started turning up out past Pluto. Some of them were spherical, like a planet. So, were they planets?
If you take a look back at Pluto's discovery, you can make an argument that the only reason Pluto was called a planet from the start was because Tombaugh was trying to find a planet. He found something, there's no doubt there. But would it have been called a planet if they weren't specifically looking for a planet? As it turns out, Pluto's mass (including the whole Pluto/Charon system which includes four other "moons") isn't nearly enough to explain the discrepancies between the observed and calculated positions of Neptune (which, as it happens, were later explained away by calculations based on Einstein's discovery of special relativity).
When Mike Brown discovered something that was potentially larger than Pluto, but much further out, and just as weird, people started questioning what it meant. If Eris was a planet, then what made it a planet and not Sedna, or Orcus, Or MakeMake, or the others. And then, if those were planets, why not Ceres, which had been downgraded to an asteroid. What does "planet" mean?
So here come the semantic police. The problem is, who decides what a word means. We run into this in language all the time. But for science, we need precise definitions so we can make and categorize observations. If I simply define a star as a bright light in the sky, that includes planets, the moon, and even airplanes. For science to work, we have to have agreed upon definitions - operational definitions.
Of course, in many cases, there's no single body to define things. There is no "official" body to make these decisions. The purpose of the International Astronomical Union is to provide an agreed-upon group to set standards like this. When they met in 2006, they had a working group to formulate a definition of what should define a planet and differentiate it from another body. To some greater or lesser extent, it IS arbitrary - as is all language.
When I say fruit, you might think "banana" or "apple." But not everyone would think of "tomato" as a fruit, even though it is, by definition. How about a cucumber? Both a tomato and a cucumber are the fruiting bodies of their plants. They are, by definition, fruits. Yet how many people do you know that talk of a cucumber as a fruit? Most people consider fruits to be sweet while vegetables are savory. So, to an extent, it will depend on your point of view.
But for science, we need a standardized set of definitions. And this is what the IAU did in 2006. Their working group formulated a set of criteria to define a planet, and put it to a vote of the general assembly, which passed it.
The authors of the cited article state that "taxonomical terms like planet are determined by the scientific process, not voting." However, when you are arguing semantics, then you are talking about consensus. Looking at another taxonomic system, scientists agree that a "cat" is a cat because it has certain characteristics. More recently, those characteristics include genetics and assumed evolution, but historically taxonomy was more based on observed characteristics. In defining a planet, we have the same process at work - only the defining characteristics were put to a vote.
Regardless of what we call Pluto, no one argues that it is not a definite body of some substantial size orbiting the star Sol. The argument does not deny Pluto's existence, only its classification. Pluto, along with the eight other fully-accepted planets, along with the thousands of asteroids, along with the myriad other trans-Neptunian objects, Plutinos, Kuiper Belt objects, and everything else orbiting the sun, are all planet-like in some ways, and not in others. At some point, a definition must be agreed upon, and this is exactly what the IAU did in 2006. Not everyone agreed, but that wasn't necessary. We weren't stating a scientific fact, we were assigning a label. It's not a FACT that anything is a planet, it is simply a label of convenience. In fact, if you wanted to be nit-picky, you could argue that Jupiter and Earth are not at all the same type of object, and that they shouldn't both be called planets but should have different classifications.
By reclassifying Pluto, the IAU basically played a game of "one of these things is not like the others" and drew a line. And whether or not you agree with that line, the majority of the assembled members of the convention did, and adopted the definition. Sure, it could be changed, but why bother? It really makes no difference other than in the minds of people who don't understand otherwise. It's PR, and nothing more.