Under the perennials section at your Garden Guide link for cleaning up for fall, it states that hay can be used for mulch. Isn’t hay used for horses, and straw for mulch?
Either straw or hay can be used for mulch, with straw often being used if the goal is simply to cover the soil, while hay will enrich it.
Straw is the stalks that remain after grains have been harvested. It is dry, hollow and lacks nutrients. It also contains fewer seeds than does hay.
Hay is grass that is cut while it is green & nutrient-rich, but before it goes to seed. There are always seeds present in hay, and some will likely grow to be weeds. Hay is soft, densely packed and spongy, soaking up water so that it stays moist. Unless the area is very wet, it should not mold, but will decompose into compost that provides nutrients for your plants. As well, hay acts as a buffer and helps to prevent excessive acidity/alkalinity of the soil. If you choose hay, purchase the best quality you can find, which will contain the fewest seeds.