It is normal in my home village for ponies to have different names in different places. For example, one of the old mares that used to wander around this area was called Jubilee on the lower road and known as Mary on the higher road. I'm afraid we tended to call her Wormy because she was always very lean (sorry!). In our early days here, she had a foal in our garden; she gave everything to her foals.
Later she had a foal which was sold to one of the people who lived in the village who fancied being a Commoner. However, he kept Rumball, as he called him, in the garage and got very cross when he walked on the lawn! When he left the New Forest, he gave Rumball, who we knew as Peggly, to my friend Angelika.