Privateers are officially chartered by the crown. They are not pirates. There's a world of difference. Privateers were often funded by businessmen looking to turn a profit. That's really the point of privateers, to get someone other than the crown to pay for it.
Now to the people on the other side of the conflict, there's really not much difference between privateers and pirates. Except that privateers are probably better funded and better equipped, so more dangerous. But still, they do the same thing.
I think that Corsair is just the French version of privateers. But it may be a more accurate term. According to my quick check of Wikipedia privateer is a more modern term, started somewhere around the 1600s, while corsair is much older. Or maybe I'm just reading something into it that isn't there.
Anyway, "pirate" is the completely wrong term. Pirates are outlaws, subject to being hung by whoever catches them. If you're a realm, and your realm engages in a certain practice, it's not illegal in your realm. So you can't be a pirate, because piracy is an illegal act. From your own point of view, it's commerce raiding, privateering, or legitimate warfare. It may be semantics, but it an important distinction.
