I never knew there was such a thing as "horse hay" or "cow hay" until the past year when I started boarding off the farm.
Around the farm I was used to farmers being able to tell hay and hay was sold according to how much rain that season brought and how long it sat in the field.
Last winter I had to buy some hay of a guy for 8$ a bale. The know-it-all horse lady of the barn bought the same hay from the same guy for 13$. He then explained to me on a later date that horse people stand out and will fork out the extra cash for just the title of "horse hay".
What I was feeding my mare was apparently crap "cow hay" from a ditch. It was 40% alfalfa and 60% blue, brome and local native grass mix, baled the week before she saw it. She then proceeded to walk over to the bale that a boarder and I had just pulled over to the junk pile. THAT was horse hay. It was 100% native grass and over dried that she was just using as a filler for her overweight horse to have something to munch on longer. My mare would shrivel up and be a rack of bones even if she had 24/7 access to it.
A city fellow moved out by parents and figured he'd start up a hay farm. His "horse hay" is 80% weeds and 20% garbage off the highway. He baled it while it was raining and moved it directly into a completely closed building. He still thinks he can get 15$ a bale (you can find hay bales that size around here for 7-12$, depending on who and how much alfalfa).
So basically I find "horse hay" and "cow hay" are subjective to the person buying and selling.