Functionally, what’s marketed as domestic first by US airlines is really equivalent to international premium economy; at least you get business class credit for it. The “first” bonus is for international three-class (or four-class if you count premium economy) first, which is a whole different animal both in product and price.

AA used to use F fare buckets for domestic first like AS still does. (When domestic first was F, the same product on short haul international flights [eg Canada] plus Puerto Rico was marketed as business and booked into J, which was even sillier.) That was always weird since AA does offer three/four-class F. AA reassigned the domestic first cabin to J fare buckets several years ago, which IMO is a rationalization. But the marketing of that product as “domestic first” remains confusing if you don’t know better.