There are many theories and definitions of what a short story should be. However, the most common working definition relates to the story's focus: a short story narrates a single pivotal event in the lives of the characters. (It also has a setting, a point of view, a theme, etc., but those are secondary to the narration of an event that changes the characters. The choice to exchange such well-chosen gifts, and then to receive them with love (and without bitterness), is that central event in this story. It is why O. Henry can end with these lines:
"And here I have lamely related to you the uneventful chronicle of two foolish children in a flat who most unwisely sacrificed for each other the greatest treasures of their house. But in a last word to the wise of these days let it be said that of all who give gifts these two were the wisest. O all who give and receive gifts, such as they are wisest. Everywhere they are wisest. They are the magi."
Edgar Allen Poe used Aristotle's theory of unity for a play as also the criteria for a good short story. A short story should have unity of time, unity of place, and unity of action. This means that the plot will usually be fixed in one place, with the action happening over a relatively brief period of time (the setting). The story line will not be elaborated into complicated plots and subplots but will usually deal with one conflict, then focus on its resolution. Of course there are exceptions, but this is a good rule of thumb to follow.
You can see these guidelines applied in "The Gift of the Magi." The reader only sees things from Della's point of view although what happened to Jim when he made his purchase is just as important to the outcome of the story.
An example of exception is the Nebula award winner "They're Made Out of Meat" by Terry Bisson. The whole thing is just a dialogue between two extraterrestial beings. There is no setting, no action, just talk; but it's great!