It seems to fundamentally change what it means “to orbit” something.

As I understood the term, orbiting would be used correctly in these cases:

• A lighter object orbits a heavier object, and both of their paths of motion are elliptical about their barycenter

• Two objects of identical mass orbit each other, and their paths of motion are circular about their barycenter

In contrast, the image above implies the following:

• A lighter object does not orbit a heavier object; they both orbit their barycenter with an elliptical path of motion

• Two objects of identical mass do not orbit each other; they both orbit their barycenter with a circular path of motion

Even the Wikipedia page for barycenter, which OP linked to, opens with the following:

“the barycenter… is the center of mass of two or more bodies that orbit one another and is the point about which the bodies orbit.”

Perhaps “orbit” as a verb has two meanings, depending on the specificity of the context.