Since the Moon's orbit is always concave towards the Sun, one could even argue that the Moon is a planet, not a satellite, it just happens to have an orbit that is closely coupled to that of another planet, Earth.
As I understand it, in fact, there was discussion about defining systems like Earth-Moon as double planets, but that didn't make it into the final definition.
I have heard people argue that we are in fact in a kind of binary planetary system. No other planet has such a large satellite relative to its size, until you get to the likes of Pluto etc.
The Earth is closer but I don't think there's any good argument to say it's actually a binary system. Notably the Earth-Moon barycenter stays inside the Earth, and the Sun-Jupiter barycenter doesn't stay inside the Sun. But barycenters are too easy to get outside of objects anyway.
As I understand it, one suggested "double planet" definition was that the barycenter would have to be outside both objects. Earth-Moon doesn't (quite) satisfy that condition, but for example, Pluto-Charon does.
>the Moon's orbit is always concave towards the Sun
Huh?
Here’s a visualisation https://www.youtube.com/watch?v=KBcxuM-qXec
The Sun's gravitational pull on the Moon is always stronger than the Earth's. (Do the math and see--on average the Sun's pull is more than twice as strong as the Earth's.) So the net acceleration of the Moon is always towards the Sun, never towards the Earth.