To put it the other way around, that's either the position of the barycenter relative to the sun, or the position of the sun relative to the barycenter. If you're going to pick one and say it's not moving, the barycenter is a good choice. The barycenter only moves relative to the rest of the galaxy.
My understanding was the sun is orbiting around a constantly (though subtly) shifting point because of the complex interactions of all the planets that orbit it. Each planet orbiting the sun alone would have a different barycenter. Put them all together and you get a much more complex picture. Correct me if I'm wrong.
It's not that the barycenter is moving, it's that the orbit of the sun is not even remotely circular. The barycenter only moves in response to forces outside the solar system (the galaxy).
On the other hand, if you look at the barycenter of, say, Earth + Sun, then that barycenter will move around because there are a bunch of other planets involved. But the barycenter of the solar system is more stable, due to the conservation of momentum and all that, and the distance from the solar system to the nearest objects that would influence it.
