"Realism" is a very broad but useful term that can be applied to a wide range of literature. "Psychologica realism" should be, by definition, a sub-category of "realism." I'll do my best to explain the two, very briefly, and to answer your question about TIm O'Brien.
To begin with, realism has nothing to do with "what actually happens." A perfect example of realism could be completely fictional, for example, and there's even a very famous and popular sub-category of realism called "magical realism," in which angels can fall from the sky or butterflies can emerge from dying people's mouths. Instead, literary realism might be defined as an attempt through literature to capture the lived experiences of people. Realist authors pay very close attention to the environments of their characters as well as to forces (e.g. economic class or racial or gender discrimination) that limit the actions of individual charactesr in their works of literature.
Henry James is often held up as one of the primary examples of psychological realism. He is more concerned with his characters' inner worlds, but he paints those worlds as convincingly as any realist author might. I think that John Cheever's short story "The Swimmer" is a good example of psychological realism. Cheever isn't concerned with telling a purely factual story, but he does paint a convincing porrait of surburban America and the detrimental effects of excessive alcohol use.
I'm less sure about Tim O'Brien's "The Things They Carried." That story (and the collection with the same name) may fit in the category of psychological realism, too. You may want to look at the narratory's distinction in "Good Form" between "story truth" (a fictional but "true" treatment) and "happening truth" (a factual but perhaps less "true" treament of the same subject).