A planet is an object that:

  a) orbits a star, and is not a star nor a brown dwarf itself;
     i) a star is an object that is massive enough and hot enough to conduct protium fusion in its core
    ii) a brown dwarf is an object that is massive enough to conduct deuterium fusion, but insufficiently massive (and hence insufficiently hot) for protium fusion  
  b) has attained hydrostatic equilibrium, that is, it is spheroidal;
  c) it has cleared its local neighbourhood of 'rival planets' since its birth;

There is some controversy about when giant planets stop being giant planets and start being brown dwarfs—typically around 30 Jupiter masses—but this is an active area of research, to understand the fusion ignition in these objects. As I mentioned in another comment, classifications are useful. Of course they are arbitrary and they are biased—they were decided by humans. But the entire point of science is to push the edges of our understanding and reduce these biases and make our models more general and less arbitrary.

I agree with this picture as the barycenter argument seals it for me. The only leap left is for me to believe Ol'Jove as a failed star (it still emits heat in excess of what it derives from Sol) and we find ourselves cozily herded about by a binary star system.