“The earth would be a moon of Jupiter not the other way around.”
Perhaps that is one way of looking at it, but you should realize the Moon/Luna does not orbit around the Earth. The Earth does not orbit around the Moon/Luna. They both orbit around a point in space known as the barycenter, where their gravitational forces equalize. It just so happens the barycenter for the Earth and the Moon/Luna is located near the Earth’s sea level.
By definition, the moon does orbit the earth — a smaller body in orbit around a larger body. However, the barycenter is center of mass of the earth and the moon. Libration, or lagrangian, points are points in space where the gravitational forces equalize. Astronomers look for shifts in stars to help them determine if the star has any bodies orbiting it. I don’t think that anyone really says that the associated planets don’t orbit the star because the associated barycenter isn’t colocated with the star’s center of mass.