If I'm wrong, I'm sure someone will come along to correct me. But I don't think the above statement is correct. All motion is relative in inertial reference frames, but the Sun-Earth system is not an inertial reference frame due to the centripetal acceleration.
I think the only correct (physics) way to look at it is to say that the Sun and the Earth both orbit about their combined center of mass, which is located inside the Sun, because the Sun is so big.
Again, if I'm wrong I hope someone corrects me.
That at the smaller scale, but add to that the Sun is also orbiting the center of the Milky Way Galaxy, which itself is moving...
“Again, if I’m wrong I hope someone corrects me.”
You almost got it correct, because the Sun and all other bodies of mass (planetary bodies, asteroids, comets, meteors, dust gas) in the Solar System orbit around their common barycenter of mutual gravitational attraction, which happens to be located within the Sun due to the Sun’s overwhelmingly greater proportion of mass versus the other matter in the Solar System.