In addition to the two versions of the claim in the question, the claim is repeated here by Andrew Solomon. I could not find other instances of the claim online. The article seems to have been presented at the NIMH (National Institute of Mental Health) Alliance for Research Progress Meeting.
I was particularly taken by a study in which a group of depressed people and a group of nondepressed people were given a video game for an hour and then were asked how many little monsters they thought they had killed. The depressed people were by and large accurate to within eight or nine percent. And the nondepressed people guessed between eight and twelve times as many little monsters as they had actually killed (see Taylor, 1989).
In the references, "Taylor, 1989" (the cited source) is presented as:
Taylor, S. E. (1989), Positive Illusions. New York: Basic Books.
This is a reference to Positive Illusions: Creative Self-Deception and the Healthy Mind, published by Shelley Taylor. (Thanks for correcting me, @ruakh.) I have tried searching the online Google Book for various keywords, without obtaining relevant results. I have also accessed a print version of the book, and flipped around to places where I think the study might be mentioned. I have not found the study. The contents and main idea of the book do not seem that related to the results and implications of the study (that depressed people can more accurately view the world).
I also looked at Taylor's journal articles and none fit the description of the study. Taylor didn't study depression that much (if at all) and instead focused on the more social aspects of psychology. There is only one mention of "depress[ion]" in the title of her journal articles — Early family environment, current adversity, the serotonin transporter polymorphism, and depressive symptomatology (2006).
In addition, I emailed Taylor (author of the book). She did not remember this study and additionally confirmed what I suspected.
The study [you] are referring to was not done by our lab.
I am starting to doubt the source and am still waiting on a reply by Solomon. Why not cite the study or the author of the study? Why cite a 301 page book instead? So that it's almost impossible to eliminate this book as the original source? Why aren't there more stories of this study online?
Did depressed people far more accurately estimate how many monsters they killed in a video game?
What I know so far is the author of the cited source doesn't remember and didn't perform the study. I also cannot directly confirm the source from the person making the claim.
The previous version of the answer incorrectly focused on the WHO.