Setting the line between "planet" and "dwarf planet" based on whether a body has cleared the neighborhood around its orbit is understandable. It reflects our understanding of how solar systems form, and it can be numerically defined to a reasonable degree of specificity. However, it's kind of arbitrary, and it's not always easy to explain to laypeople. Moreover, it's somewhat unstable; if there were slight differences in our solar system, it wouldn't work. For example, if Saturn and Jupiter were in a true resonance, would this mean Saturn wasn't actually a planet because it was part of Jupiter's influence? Or if Mars moved outward and began clearing portions of the asteroid belt, would it cease to be a planet until after it had cleared the asteroid belt, then resume planetary status? The Stern–Levison parameter Λ has a strong dependence on orbital distance; if Mercury had a perfectly stable orbit at the average orbital radius of Eris, it would stop being a planet and start being a dwarf planet. How might this definition apply to other stellar systems? Even though we can come up with a numerical discriminant to define clearing one's orbital neighborhood, it's not easily represented in a concrete way.
I submit that when laypeople use the term "planet", the primary referent is to a place in a hierarchy. Most people with a passing knowledge of astronomy understand that stars, planets, moons, asteroids, and comets are all different things, occupying different "slots" in the solar system. Thus, the word "planet" is defined in common usage within the context of stars, planets, asteroids, and comets. I'd like to come up with a better definition that reflects this usage without sacrificing scientific functionality or rigor.
When people think about "moons", they typically don't think about small, irregular rubble piles like Phobos or (66391) 1999 KW4; they're thinking about objects like our own moon. In public imagination, a "moon" is itself a gravitationally rounded object around a planet. And this, I think, can give rise to a better definition of "planet".
We ought to define a planet as any body in its own solar orbit which is massive enough to retain a stable gravitationally-rounded satellite of comparable density. Obviously, a planet doesn't need to have round moon; if that was the case, Mars, Venus, and Mercury would all fail. But Mars, Venus, and Mercury are all more than large enough that they could retain such a moon absent outside gravitational perturbation.
The largest known body which is not gravitationally rounded is the icy moon Iapetus, with an average radius of 735 km, and it would definitely be gravitationally rounded if its density was higher. An average radius of 735 km is therefore a good minimum estimate for gravitationally-rounded bodies. In order to be considered a stable satellite, the primary must be large enough to have a barycentre contained within itself at a distance large enough that the satellite is not in danger of being torn apart by the tidal gradient.
For the larger planets, this isn't an issue; a moon with a radius of 735 km and the same density as Earth would need to be over 600 Earth radii away before the barycentre would be at the Earth's surface, while the Roche limit for a secondary of equal density to its primary is just 2.455 Earth radii. But for Mercury, it would be just 37 radii, and for Eris or Pluto it would be around 4 radii. Bodies with a diameter greater than 1,851 km can stably retain a gravitationally-rounded moon of comparable density and should be considered planets; bodies with a lower diameter cannot.
Does this mean Eris and Pluto should be considered planets? Well...almost. There's one other aspect of what defines a "planet" in the common mind, and that's the neat, circular, nested orbital pattern depicted in solar system models:
Now, obviously we shouldn't be reinforcing this view, but the idea of the planets being nested fully inside each other's orbits is pretty firmly fixed in the public's mind. And in terms of solar system evolution, a body large enough to clear its own orbit will have also settled into this kind of orbit. Indeed, this image provides a fairly picture clear of a body's "neighborhood", where you start in Mercury's neighborhood and then cross into Venus's, Earth's, Mars's, Jupiter's, and so forth as you move away from the Sun.
Bodies that are otherwise large enough to be planets but with orbits that share part of the orbit of another less eccentric planet-sized body, then, shouldn't be considered planets. Pluto crosses the orbit of Neptune, so it's out. Eris crosses the orbit of Pluto, so it is also out. Ceres does not cross the orbit of a less eccentric body, but it is not large enough to retain a gravitationally-rounded moon, so it's not a planet either. Incidentally, this also defines why moons aren't considered planets: even though the largest ones could be considered planets if they were on their own, they continually cross the orbit of their primary.
This would make it so much easier to explain why Pluto isn't a planet. "In order for something to be considered a planet, it needs to have its own orbit and not cross into the territory of other planets. It also needs to be big enough to have moons of its own...a little over half the size of our moon." Simple.