Modern History of the term "Planet"
It is actually not the first time in astronomical history that objects were first included in, and then later excluded from planetary status. At the turn of the 19th Century, after pondering the apparent "gap" in Orbit #5 of Bode's Law (Bode's Law actually turns out to be just a coincidence, not a law), people started looking for an "8th" planet in this apparent gap between Mars and Jupiter. After intense searching, a small body about 500 miles in diameter was indeed discovered and named "Ceres", bringing the number of known planets up to 8: Mercury Venus, Earth, Mars, Ceres, Jupiter, Saturn, and Uranus (all fitting Bode's Law quite nicely). Over the next several years, 3 more planets were discovered in this region, which were subsequently named: Vesta, Pallas, and Juno, bringing the number of Planets up to 11. In 1846, Neptune was discovered (mysteriously "breaking" Bode's Law), nominally bringing the number of planets up to 12. However, in the meantime many more smaller bodies had been discovered orbiting between Mars and Jupiter, potentially increasing the number of planets in the Solar System to as high as 50. It was soon realized that calling every little chunk of rock that they found (and that is what many of them were) a planet was not only going to become cumbersome, but also break the unspoken and undefined sense of what a planet was supposed to be. Hence the terms asteroid and planetoid came into being, and everything in the newly termed "Asteroid Belt" was demoted from planetary status (lowering the number of planets in the Solar System back to 8).
With Pluto, you have almost the exact same situation occurring. Pluto was originally discovered by itself in 1930 and believed to be larger than Earth. Over time, revised estimates reduced its size to about that of the planet Mars. In the late 1970's, better imaging discovered not only that Pluto was much smaller than Mars (it was even smaller than Mercury), but that part of its apparent "size" was due to a heretofore unknown large and close-orbiting moon (dubbed "Charon"). By the 1980's it was theorized (and over the next 20 years verified) that in the region beyond Pluto there should be a belt of icy bodies in a fairly wide band that are ultimately responsible for many short-period comets. This belt has been named the Kuiper Belt. It turns out, just as with the Asteroid Belt about 200 years prior, Pluto is simply a large member orbiting along the inner edge of that belt (there are some other Kuiper Belt Objects [KBOs] that are comparable in size and at least one, Eris, that is apparently larger than Pluto). So the question became (just as with Ceres, Vesta, etc. in the the Asteroid Belt), "What makes Pluto so special that it should be defined as a planet, but all of the comparable bodies in the Belt not planets?" But it was also realized that self-rounded bodies due to self-gravitation were different from floating boulders, ice-chunks, and rubble piles, so a new category "Dwarf Planets" was created.
Personally, I would have preferred that the existing term "planetoid" be used and modified (further distinguishing the term from the more generic "asteroid"), but that is just me.
Ultimately the issue is that we all have a "sense" of what a planet is, but if we were asked to rigorously define the term (as is normally done in scientific circles), we would stumble around coming up with a clear definition. It was the Sci-fi author James Blish who invented the term "Gas Giant" in the 1950's. Before that, Jupiter was a "planet" no different from Earth or Mars being a "planet". But with the term "Gas Giant", a meaningful and descriptive subcategory was created to distinguish one class of planet from another. Thus, we now have, Gas Giants, Rocky (Terrestrial) Planets, and Dwarf Planets. And in exoplanet terminology, Gas Giants are now being subdivided as well into "Jovian" versus "Neptunes" (i.e. Gas Giant vs. Ice-Giant).
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