Short answer: no, a short story does not need a definitive ending (for suitable definitions of the term definitive ending.)
Longer answer: Short stories can be classified and analyzed and grouped in all sorts of different ways but they can be loosely divided into literary and genre stories (with lots of caveats and disagreements and gray areas.) This division may also loosely correlate with character driven and plot driven stories.
Examples of literary stories are works like <PERSON>'s The Snows of Kilimanjaro, <PERSON>'s Gimpel the Fool and <PERSON>'s An Occurrence at Owl Creek Bridge. (You can likely find all of these available online if you're not familiar with them.) They often do not have a definitive ending, and it may be difficult to articulate their plot in terms of conflict and resolution. Some people (particularly long-suffering high school students forced to analyze them for a graded essay) may claim they don't have a plot at all. Their intent is generally not to tell a recognizable story. Their purpose is "... that the reader shall come away with the satisfactory feeling that a particular insight into human character has been gained, or that his (or her) knowledge of life has been deepened, or that pity, love or sympathy for a human being is awakened." <PERSON>Genre stories, on the other hand, tend to be more traditional tales. 460
They may be detective stories or romances or fantasies or any of the dozens of other genres that we use to distinguish between types of books. And like any other work of a given type, they have tropes and expectations that go along with the genre. For most if not all of these, the reader expects a definitive ending. They expect a conflict and a resolution. If you present your story as a genre story and then fail to meet the reader's expectations, you will generally engender frustration and dislike. That doesn't mean you can't do it but it does mean that it is much more difficult to make your story successful in the reader's eyes. If, as you suggest, you're writing more for your own amusement than any expectation of commercial or external success, then of course you can do anything you like. 846