Son of a farmer, living in a farmer community. Alfalfa hay doesn't have seeds unless you harvested it waaay too late, and then it's probably worthless.
I've only ever heard of straw referring to wheat straw - the wheat stems (and grainless heads) left in the field after harvesting.
Hay, on the other hand, is the whole plant (well, not the roots) of alfalfa (a legume), or brome (a grass) or prairie hay (mix of native grasses).
This may be slightly different in other areas, but the above is how it is in Kansas.
You can cover your new grass seed with straw to protect it from birds. If you try that with hay you'll plant hay.