To me, hay is has not been threshed and is harvested with the thought of retaining the seeds. Straw has been threshed. Threshing removes most seeds but not necessarily all. How much seeds comes out depends on variety and species, how ripe it is, and probably how moist it is. I've never been around true threshing for harvest, just at fairs and exhibitions. I imagine you can find straw for about every variety of grain that is grown somewhere in the world. I've heard that rice straw is fairly common some places in the southeast.
Seedcorn, in East Tennessee Dad's first hay cutting was pretty much Orchard Grass. We only got two cuttings a year. The second cutting was either Lespedeza and Red Clover or Fescue and Red Clover or sometimes Lespadeza, Fescus and Red Clover. We fed both to the plow horses and cows.
One of the neighbors would even bail soybeans. I hated those. The soybeans never really dried out enough to bale and he would have to bust the bails apart when they got to the barn or they would get really hot. The concern was spontaneous combustion. The reasons I hated them was that the bales would be really heavy and hard to handle since they were wet and the fine dry leaves would get all over you and really itch. Still I needed that $0.50 per hour he paid me to help get his hay up so I went after that job.
I also don't like oat straw, but not for your reason. That stuff is slick! On year Dad grew oats. We put it in the barn as loose hay, not threshed and not baled. When we hauled loose hay, my job was to stack it on the horse-drawn hay wagon. Dad threw it up on the wagon with a pitchfork and I spread it out. The only load of loose hay I ever lost any of was one of those loads of oats.
I was probably 11 or 12 so that was many decades ago, but I'm still defensive about that. Oats are slick, it was going uphill, and Dad jerked the wagon when he started the horses, but it was my job to stack it so it did not come off. I failed.