Is it a "Bary center" or a "Bari center"?
It is safe to say that Barycenter is the correct spelling for the sought word. The other versions are artifact of the original word as some authors preferred to name (especially in the internet).
Barycenter (or Barycentre in British English) is a commonly used term in astronomy and in chemistry (in Crystal Field Theory) as well. In both fields, it is used to explain center of mass balance and most of the literature has used the aforementioned spelling. For example, NASA explains the question &quot;What Is a Barycenter?&quot; as:

We say that planets orbit stars, but that’s not the whole truth. Planets and stars actually orbit around their common center of mass. This common center of mass is called the barycenter. Barycenters also help astronomers search for planets beyond our solar system!

In Crystal Field Theory (CFT), electrons in the $\mathrm{d}$-orbitals are repelled to different degrees by the ligands. The degeneracy of the $\mathrm{d}$-orbitals is lifted. The Barycenter is the average energy of the $\mathrm{d}$-orbitals (Coordination Chemistry II: Theories of Electronic Structure and Crystal Field Theory):