The NY Times, as far as I know, invented the idea of a public editor, so I suppose it's up to them to decide what one is. That would be fair. But life isn't fair.
A public editor should be, imho, the representative of The Public, on the payroll of a news organization. The editor should have the ability to publish stories alongside news reports, with equal prominence, without editorial interference or oversight. No review. Perhaps some space limitations, that seems reasonable, but no one filtering what they write.
This person must not identify with the people who write the news. He or she should probably not even work in the same office. Should not go to lunch with them. It should be impossible for them to be promoted to a news function. This is an outsider's job.
Instead of "All the news that's fit to print" the Public Editor's motto is "I bite the hand that feeds me."
The job of the Public Editor is to challenge the news function when they report as fact things that they don't provide any evidence of. To challenge the news function when it does He Said/She Said reporting, on questions that are not in dispute. Or when they fail to present a legitimate point of view, or relevant facts because they would anger a faithful source, or would buck conventional wisdom. Or would ask, for example, why tech reporters report glowingly about companies that might employ a tech reporter after they are laid off by a publication such as the NY Times.
The Public Editor would report when one of the staffers goes to work for someone they previously covered, especially if the coverage was favorable. That would be big news.
It must be someone who regularly reads the stories and yells at the screen that these people are idiots or assholes or sold out.
Reporters should hear the voice of the Public Editor as they're writing their stories. It's okay, even great, if they hate that voice. But they must anticipate it. This is where the Public gets the benefit from the existence of the Public Editor.
The Public Editor runs a linkblog that points to others who are critical of the news organization they work for.
A good Public Editor is over-the-top critical of the news organization. He or she errs on the side of being fair to the Public and unfair to the news organization.
The Public Editors the Times has hired have flipped it the other way around. They are way too understanding of the foibles of a professional news organization. And they have a career path that prevents them from saying anything too controversial. As a result, the Public Editors have basically been Seat Warmers. Their job seems to be to make the news people feel good about themselves, which is a poor excuse for a job in a news industry that's struggling to stay afloat.