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.
“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.”
When you are talking about a binary star system where the two stars are nearly equal in mass, the barycenter is going to be located in the middle of space between them and not within the mass of either, so it is hardly accurate to describe Star A as orbiting around Star B or vice versa. The reality is that they are indeed orbiting around their common barycenter in space. Likewise for twin planets of equal mass, they too will orbit around a common barycenter located well beyond any atmospheres they may have. While most of these natural satellites are typically going to be mismatched in mass to the point the barycenter will be located within one of the two masses, it is useful to remember they are really orbiting around the barycenter, which can influence many of the diurnal and nutation characteristics of the primary mass where the barycenter may be found.