Sometimes those details seem narcissistic when written in first person; the character appears self-absorbed and not really aware of what anyone else is thinking. When the writer writes in third person, though, we readers suspect, "if the narrator thinks there is something in this dreary creature worth writing about there must be." If the writer cares, then maybe so will we.
While the essay begins with this note, it slyly continues as a peek into the writer's creative process. Berlin builds a character, "Henrietta," as the example, then spins Henrietta into motion. While she does this, she also lets us know what parts of her own life she is including, what she is assembling, and how she makes a composite story. In short: how fiction and nonfiction merge for her. The story (and the essay does become a story) unfolds easily as days of the week, with a little flip at the end. All this in a compelling and compact four pages.
What is curious is that the stories up to this point in the book are written in first person, as are most that follow. The difference, perhaps, is that many strange things happen, and the stakes are high.
Maybe your story is true. Try stepping back and rewriting it. Get some distance. In the Foreward by Lydia Davis, Berlin (xvii) is quoted that some of her more difficult stories are "A transformation, not a distortion of the truth." Edit and shape to fit. Sometimes by altering something you can get to the heart of the story with more clarity and feeling. And that's what we're trying to do here, right?
One about fiction/nonfiction is here.
Another fiction/nonfiction is here.
Stakes of a story exercise is here.