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"Hay" usually refers to alfalfa but can also (and more correctly) be wild grasses which have been allowed to grow, then cut, let dry, and bailed, or even wheat or oats which have been cut with the grain still attached, allowed to dry, then bailed. "Straw" on the other hand, refers to the "stalks" of wheat, oats, etc., which are usually "spit out" the back of the combine (harvesting machine) after the grain has been harvested. These stalks are then bailed.
FWIW my mother always used straw but, as others have noted, there are as many opinions as options as to which is best.
While oat hay is excellent feed when it has been cured and handled correctly, if it has been bailed while it was too green it can develop a very nasty strain of mold which will abort calves in pregnant cows or, if the cow gets enough, I've seen them die from it. Not sure how chickens would react to it but I'm betting not good.
Sorry if this is way more than you wanted to know.