Not meaning to cinfuse the issue, here is my two cents worth.
Although it could be a privateer, the American Heritage dictionary refers to it as a pirate like that used on the Barbary coast (North Africa) or a privately sanctioned pirate. There is no mention of the word privateer.
In my years of reading and researching maritime history, I have never heard the word corsair widely applied to privateers. Usually, when one sees the word privateer, it refers to a government sanctioned, privately owned ship operating in favor of whatever government sanctioned its use.
Corsair is most often associated with pirates. A pirate operates without any such government sanction. The word corsair has roots in Latin, French, and Italian that all refer to pirates.
However, it may be that one man's pirate is another man's privateer, so the issue can be argued either way without much chance of anything being decided. I'll leave it there. :)
Russ