Doesn't really matter what you call it. "Staysail" is close enough. Beyond that, who cares? They don't make much difference anyway except to kill your view forward. Though if you blow out the spinnaker, or lose the halyard, they're at least useful in giving you some sail forward of the CLR so you can bear off if you have to.
The boat in the photo is at hull speed. Doubtful the (whatever you want to call it) is adding anything.