It seems where I live alot of people buy the name, 'horse hay' for horses must be better quality and 'cow hay' must be poorer quality, my other half is a farmer and there is no difference as to the hay quality that they feed the cows or sell to livery yards in the area. Too many people seem to assume that the hay is different when it is usually from the same field, the same cut and is wrapped at the same time. Many times I have gone down the farm and the hay they are feeding the cows seems to be of better quality than the bale of hay my horse was eating! I don't think its cows that will eat anything, rather that farmers will give the cows a bale that horse owners would not think suitable for their horse.
However sometimes the horse may require different qualities... the good-doer will need the bulk rather than the nutrients from the hay, whereas for a poor-doer they will need good quality forage to supply their nutrient requirements.