I don't think anyone is suggesting you don't help an animal in distress. We would all help a horse or goat with its leg caught in the fence.
But as far as the other matters go, yes, it is just a difference in opinion on care. Routine vet care is absolutely their call. If I had a large field with enough pasture to keep the horses well fed, why would I feed hay at the same time? They generally prefer fresh grass anyway.
As far as grain goes, at my barn there are 8 horses, and only 5 get grain, the seniors and the ones in competition. The rest don't need it, they have excellent weight and health on hay alone. That is just how my BO does it, I have no negative opinions on whether one wants to feed a horse grain or not, so long as they are healthy. So, you are not obligated to feed this other horse, he is probably just fine without it. -
You can just bring yours in to feed, it won't hurt the other horse to be lonely for 10 min. If you don't want to bring him all the way in, then bring a bucket with you, and bring your horse just outside the pasture and hold him while he eats. -
You definitely should not be riding in the same field as the other horse, esp since he sounds pushy, that could be dangerous. Instead of moving the other horse, why don't you just ride in the other field? Leave the other horse where he was, so you can't get blamed if he hurts himself.-
I know it is hard when you think a horse isn't getting the same standard of care that your horse gets, but as long as he is at a decent weight, he is fine.-
And when the neighbors comment on how well your horse rides and behaves compared to theirs, well, take it as the compliment that it is. Sounds like you are doing a great job with your horse.