We would mostly just say “pen.” If we had to specify, it would be “ballpoint pen,” including the “pen.” I don’t think I’ve ever heard “ballpoint” used as a noun

We would mostly just say “pen.” If we had to specify, it would be “ballpoint pen,” including the “pen.” I don’t think I’ve ever heard “ballpoint” used as a noun. It would be like saying “fountain” for “fountain pen.”