Does that mean they won't publish something that has a source that the writer doesn't know the name, or that they won't publish it if the source wants to be anonymous?
Unnamed normally refers to a source that wants to be anonymous.
Most of the time publications will avoid these, as it can set the institution up for libel.
(I did pay attention in my journalism courses that I took for my minor in college)
It really depends on what information is being released by the source and the impact it will have.
The last thing an institution wants is a defamation suit over unnamed sources.
The use of unnamed sources has nothing to do with libel or defamation lawsuits. It has to do with credibility. If the individual wants to be anonymous or nameless, the news organization is hesitant to risk their reputation. The news organization will then seek to obtain collaborating evidence to support the statements. That is why you very seldom read or hear about an anonymous source, it is usually about anonymous sources. The organization cannot act in reckless disregard of the truth. They verified the truth of the anonymous source’s statement. It may be through other unnamed sources but they did their research.
New York Times Co. v. Sullivan, 376 U.S. 254 (1964), was the United States Supreme Court case which established the actual malice standard which has to be met before press reports about public officials or public figures can be considered to be defamation and libel. It is one of the key decisions supporting the freedom of the press. The actual malice standard requires that the plaintiff in a defamation or libel case prove that the publisher of the statement in question knew that the statement was false or acted in reckless disregard of its truth or falsity. Because of the extremely high burden of proof on the plaintiff, and the difficulty in proving essentially what is inside a person's head, such cases—when they involve public figures—rarely prevail.
Any coach at Iowa or athletic team member is a public figure.