Earth is obviously a planet. Big, roughly spherical, orbits the sun. Classic planet. But, the official definition of a planet is (according to the people who define these things) a celestial body which:

  1. is in orbit around the Sun,
  2. has sufficient mass to assume hydrostatic equilibrium (a nearly round shape), and
  3. has "cleared the neighborhood" around its orbit.

This, notably, excludes Pluto, as well as exoplanets (but of course we have the term exoplanet, so I suppose that's okay). But it also excludes Earth (despite being explicitly in the list of planets later in the same document).

Celestial is a fairly specific term – a not-uncommon definition can be found here, saying: "pertaining to the sky or visible heaven, or to the universe beyond the earth’s atmosphere, as in celestial body."

Earth is non-celestial. So it doesn't fit the scope of bodies described before the numbered list.

You might claim the explicit inclusion of Earth in the list given later in the document would override this objection. But if the list was more definitive than the definition, no new planets could be discovered by definition. This is wholly unsatisfying – at that point, there's no purpose for a definition. The truth would be, planets are things sort of like this, as approved by the International Astronomical Union. So the definition takes precedence, and Earth is disqualified.

Of course, they ought to change it to "astronomical objects". Some people suggest this is a synonym for "celestial objects" despite "astronomical" not being a synonym for "celestial". I am skeptical of this – it sounds like someone saying whole numbers and natural numbers are the same thing (this analogy is best understood when referring to whole numbers being non-negative integers and natural numbers being positive integers – but the two sets are different, even when they describe different sets in different regions, as far as I can tell). They are very close to being the same, which is enough to qualify as a synonym – for instance, close is given as a synonym of precise, when, while close, they are not precisely the same thing. We should be a bit more careful in scientific wording.