Earth has one natural satellite. Its name is Moon (with a capital M).
The word moon (small m) is a simile, used for objects that are similar to the Moon. At first, it was used for anything in orbit around Earth (for example, the early artificial satellites were called "artificial moons").
The word is used for natural satellites in orbit around a planet. For example, the moons of Jupiter (when Galileo discovered them, he called them planets, because of the definition of the word "planet" back then).
Cruithne is not a "moon" of Earth. It is in orbit around the Sun. It has an orbit that is somewhat sychronised with Earth (it is said to be in resonance), but since it does not move around Earth, it is not a satellite.
If you were to draw its orbit using Earth as the reference, the orbit of Cruithne would appear to trace out a horseshoe shape, with the Sun at the centre of the horseshoe.
There are at least five such "quasi-satellites" that are big enough to be called quasi-satellites.They are not in orbit around the Earth (except, in one case, just for a few orbits every once in a while_; therefore, theya re not "moons".
And, of couse, none of them is The Moon.