It's quite hilarious the different attitudes about what makes good horse hay or good cow hay.
As far as hay storage that is dependent on many different variables also.
I've seen hay stacked outside in the western states that stays in fairly good shape except for the top and bottom tiers.
The top from weather the bottom from ground moisture getting pulled up into the dry hay.
Round bales have been left every where from in the field where they were baled right where they were ejected or carried over and placed in rows or stacked on the edges of the fields.
I've had numerous people try and tell me that net wrapped bales can be stored outside in the weather and be fine.
All I'll say about outside stored hay is that if it's on the ground it will pull moisture from the ground requardless if it's small squares, big squares, round bales or even stacked unbaled hay. You can get by in arid locations with minimal looses.
If it is done in the East you will have considerable spoilage and loose.
I cringe whenever I see round bales left out in the weather around here, all that work and money to throw 50% away.
We store many round bales outside ourselfs with one "small" difference every bale stored outside is wrapped in plastic,
we store dry rounds wrapped in plastic when space is used up. Silage round bales are all stored outside in stacked rows of big marshmallows. Just waiting to be picked up with the squeeze grapple and feed out to the cows.
As far as hay quality in my mind dairy cows and feeder beef all need high protein, energy and digestibility. Beef brood cows
need a more fiber and lots less protein. Horse hay to me just good soft grasses with minimal dust and no mold. (it's just a horse).