I think I understand now, but correct me if I'm wrong:
So if someone asked something like "do you like your horse or their horse better?" one would use "vår hest", emphasizing that we like OUR horse, specifically?
And inversely, "hesten vår" if someone asked "do you like your horse?" and responding as a general "yes, we like our horse".
Or am I just overthinking it, as usual? :P