Using your explanation then, it appears that in the galut Sim Shalom should also be said at Shabbat Minchah though I have never seen it in a siddur in the Galut only in the my siddurim from Israel. Is there a difference in the halachah/minhah between the galut and Eretz Yisroel? Thank you.
To the best of my knowledge the issue of Shalom Rav or Sim Shalom at Mincha in not dependent upon Eretz yisrael or Galut but on of difference between minhag ashkenaz or minhag Ha'Ari z'l, better known as nusach sfard. The Ashkenazi minhag based upon the Hagahot Maimoniot who I mentioned in my previous answer, and as codified in the Shulchan Aruch by the Rema 127: 2, is to say Shalom Rav at mincha, if it is not a fast day. The minhag of the Ari z"l is to say Sim Shalom at all Tefilot.. (Ba'er Hetev 127: 7) .