A celestial body is an object in our heavens, and therefore the Earth itself is not a celestial body. (See, for example, the Outer Space Treaty.)

The IAU's definitions of planet and dwarf planet refer to 'a celestial body' orbiting the Sun. These therefore do not include Earth.

The IAU then says that all other objects except satellites orbiting the Sun (which therefore includes the Earth) shall be referred to collectively as "Small Solar-System Bodies".

Obviously, I've pointed out the above somewhat tongue in cheek. No-one seriously doubts the Earth is a planet. I assume the astronomers in the IAU forgot that celestial body does not include Earth - they should have used the term astronomical body - probably because they rarely think about the Earth in their professional work. It does show how careful you need to be with language in coming up with these definitions. It's open to the IAU to redefine the term 'celestial body' for their purposes, but I'm not aware of them having done so.