Yes, Hay is feed, straw is bedding, Straw tends to be yellow and large, while Hay tends to be thinner, longer and green. We have both redily available in our feed stores, and from local farmers.Hay is dried grass, greens, clover, alfalfa, etc that is intended for ingestion.
Straw is the cut and dried stalks of wheat/barley. It's used for bedding, not ingestion. It's not easily available where I'm at, either.
Using hay in the garden is a good way of winding up with unwanted seeds in your garden, btw.