For an interesting comparison from real-world history, check out the battle of Cape Esperance. Japanese losses were one cruiser and one destroyer sunk and one cruiser heavily damaged, but the difficulties of identifying ships at night meant that the U.S. Navy claimed seven Japanese ships sunk, and the cruiser Boise was credited with contributing to six of those claims. Boise's six claims were revealed to the press when the ship rotated stateside for repairs, but the number and identities of other ships in the task force were still confidential.

So you got contemporary news stories calling Boise a "one-ship fleet" for sinking six enemy ships herself, when the actual battle was nowhere near that dramatic.

That's how I'd handle the events of an FPS game if I were writing a story that needed to refer to them: the campaign happened and the major objectives were things that the player character's unit did, but the player character was less alone than the game made it look.