It's funny because around here cow hay means two different things - either it's dairy cow hay, and it's SUCH rich alfalfa it's unsuitable except in very tiny portions for horses or it's just "filler" hay.
I don't think they actually mean it's "cow-hay" - people aren't exclusively feeding their cows garbage hay. What they mean is that cows CAN eat it whereas horses should NOT eat it. You can put half good hay and half crap hay in front of a cow and at the end of the day it's all gone - they may eat the good stuff first, but they'll eat it all period and not be bothered. Most people I know use lesser quality "filler" hay.
Anyway, also keep in mind there's no reason why she can't eat oat straw either. It has virtually no nutritional value, but is usually clean and dry. You can supplement her on alfalfa pellets/beet pulp/etc. for actual nutrition, but let her eat the oat straw just to fill her belly.
I have done it in a pinch. It's not great, but it works. When I had to, I used extra alfalfa pellets to supplement better nutrients and protein than what the hay provided. You should beed 3-5 lbs of alfalfa when you're supplementing with it.
Or, if you can find them, Timothy Alfalfa blend cubes or pellets work even better. Lower protein, but still high in nutrition. If you're worried about choke, soak the pellets or cubes with just enough water for them to break apart.
Also, watch for foxtails. Some people don't have an issue, but the barn where I used to ride had a BIG problem last year. The horses started getting sores in their mouth, colic, etc.. they couldn't figure out what was wrong. 2 horses ended up dying of colic. After that, the USDA came out. Took one look at the hay, held up a foxtail, and said 'there you go.' The horses had to have their mouths washed out.. had to have special mashes daily. After 2 weeks on good hay, their mouths healed, and they could be ridden. It was scary, though.
Around here, the serious dairy/beef folks do indeed feed high quality hay. However, excluding that industry, any hay that horse folks won't buy is commonly called 'cow hay'. That can include anything from an unmanaged field (no weed killer or fertilizer) to an unfortuneate rain on 'good' hay that was cut but not yet cured and baled.