So, is there a non-arbitrary definition that'd include the 8 and exclude the rest?

See, that's the trouble, it's a false dichotomy, really. There are all these objects that orbit the sun and they have a wide range of sizes and compositions and orbital parameters.

It would make sense to impose some kind of mass threshold, but even that would be arbitrary.

There are all sorts of interesting objects to study in the solar system. Arguing about what to call them seems kind of stupid.