I mean, the sentence either implies what I said before, or it implies that the barycenter is a point outside the sun. I really don't see any other reading than those two.

Yeah pretty much my point. I know you can maybe kinda construe it into the truth if you already know about the topic, like other commenters age saying, but it's presented as educational, and does a poor job at educating with how misleadingly it is phrased.