Let's look at some sources for the meaning of the word "Corsair".
pirate, especially : a privateer of the Barbary Coast
- a fast ship used for piracy.
- a pirate, especially formerly of the Barbary Coast.
A corsair is a privateer or pirate, especially:
- Barbary corsair, Ottoman and Berber pirates and privateers operating from North Africa
- French corsairs, privateers operating on behalf of the French crown
1 archaic A pirate.
‘Reputations spread through any community, and the pirates and corsairs knew who they wanted to work with, as well as who they did NOT want to work with…’
1.1 A pirate ship.
‘The Goshawk was probably enough to discourage most pirate corsairs from attacking, but a cruiser was another matter.’
1.2 A privateer, especially one operating along the southern coast of the Mediterranean in the 16th–18th centuries.
‘French corsairs settled on the western part of the island in the 17th century and Spain recognized the French claims to the area in 1697 in the Treaty of Ryswick.’
The Cambridge Dictionary gives:
a pirate (= person who sails in a ship and attacks other ships in order to steal from them), especially one given permission by a government to attack enemy ships in the 16th - 18th centuries in the southern Mediterranean
It would appear that the word is used to mean "pirate" as much as "privateer". It would also appear that in actual history, the targets/victims did not always clearly distinguish between pirates and privateers. The British refereed to John Paul Jones as a pirate, although he had a letter of marque from the US authorities, for example.
So far as i know there is no canon source for how Umbar was organized or governed. It is quite possible (although nowhere stated by JRRT) that the raiders were authorized by whatever leadership they had, making them technically privateers rather than pirates.
Also, please remember that the English of dialog in LOTR, including "corsairs" is only a translation from the Westron. Whether the speech of Gondor had any similar distinction between "pirate" and "privateers" is less than clear. Out-of-universe, JRRT no doubt used "Corsairs" for its somewhat antique flavor, which fit with the style of much of LOTR, particularly the scenes set in Gondor.
The distinction between "pirates" and "privateers" in historical English comes with the development of Prize Courts and other somewhat legalistic ways of dealing with sea combat and blockades during the 15th to 19th centuries. (The word "privateer" dates to the mid-17th century.) Privateers became essentially obsolete in the late 19th century, partly because the economics of commerce raiding no longer work (an effective warship is now much more expensive), partly because the Declaration of Paris bound all signatories not toi issue Letters of Marque. There have been no significant uses of privateers since the 1870 Franco-Prussian War (well before Tolkien's birth).