The embarrassing fact is that there is currently no definition of "planet." In ancient times, it was easy: a planet was one of the celestial objects that moved in predictable, well-known patterns relative to the fixed stars (which don't move relative to each other), as opposed to things like the occasional comet which were, as far as the ancients were concerned, one-shots. Later, we found out that planets are large round objects in orbit around the sun, and the thing we're standing on is a member of that category.
Things started to get messy in the 19th century with the discovery of the asteroids. Ceres was originally classed as a planet, but then was demoted when it was discovered that it was just one of many similar objects in the same general area. Pluto was discovered early in the 20th century and has been flirting with a similar fate in recent years; it is now generally regarded as a Kuiper Belt Object (KBO) and has turned out to not even be the largest of those, just the easiest to spot.
The current proposal at the international conference is to define a planet as anything massive enough to be forced into a roughly spherical shape by its own gravity, which orbits the sun. Per se, that seems to be a decent definition. If the data is available, then the criteria can be applied in an objective and deterministic way to get a definitive answer. For objects orbiting each other, there are some details about whether the center of gravity lies within one of the bodies or in free space; that could get sticky if a pair is found with very eliptical orbits, in which the center of gravity could be either above or below depending on the current positions of the bodies, but there are no known cases at the moment, and if one is found, the situation could be resolved with a minor modification to the definition.
The issue I see with the proposed definition is that it ignores certain natural groupings. Ceres would become a planet again, but the other asteroids would remain asteroids. Pluto would remain a planet, but Charon would also become a planet, as would Xena and some other KBOs. What's wrong with that? Well, consider the eight classical planets. While the four rocky planets are broadly similar to each other, as are the four gas giants, they are all isolated objects. They are not one of many in the same general area. Ceres, on the other hand, is very much like any other asteroid. And, as far as we can tell, Pluto and Charon and the other KBOs are not very different from each other. So the result of the proposed definition is that it out of groups of tens of thousands of undifferentiated objects, it picks out a handful of the largest and puts them in a different class based simply on their size.