My (utterly layman's) understanding is, basically, that precision requires some sort of definition, which planets more or less didn't have. It was a "This is a significant celestial body", without having a clear idea of what made it significant.
Eris was discovered in 2005. Because it is bigger than Pluto (about a quarter more massive), the argument went "Well, if Pluto is a planet, then so is Eris". Which led to the discussion of "Well, what IS a planet? If Pluto is a planet based on size, so should Eris be. But then what about Ceres and the larger objects in the mid-system asteroid field?"
So, they sat down and hammered out a definition. That definition excluded Pluto, Ceres, Eris, and other "dwarf planets" more or less because it did. I'd wager "We need something resembling a manageable number of planets" was part of it, but I'm not sure.