Here, cow hay is usually horse hay that has been damaged in some way. Like it got rained on, was at the bottom of the stack and has dirt or mud, some moldy areas, sometimes lots of weeds, etc.
But "cow hay" is also relative to the rest of hay the grower has. For instance, this past summer must have been a good year because my friend was buying "cow hay" in bulk for her horses, and it was GREAT hay, it was just baled a little bit dryer and fell apart easier. But there was nothing wrong with it at all. I was paying twice as much for the same quality because I had no way to buy and store it in bulk. So I can only imagine what "horse hay" for that farmer was like this past year. It must have been really pretty!