Personally, my definition of what constitutes a "short story" is pretty loose. Observe:
A short story is A) Short and B) ... nope, that's it.
By which I mean, I don't really have any expectation that a short story will give me a complete character arc or a complete "story" in the traditional sense. Many of my favorite short works serve largely to enunciate some idea, or to sketch a picture of an interaction that is more or less fully sensible without further context.
This means that a lot of my favorite short stories can easily be pieces of a larger work - which is A-OK by me. In fact, my first introduction to Ender's Game was through an excerpt included in the anthology There Will Be War. It stood perfectly well on its own.
My freely-available short "Le Morte d'Arthur" has garnered a number of reviews, most fairly positive. The most common negative mentioned, though, is that it is "too short." Now, I'm fine with this - first of all because nobody ever says a terrible story is "too short" so at least I know they wanted to read more. Second of all because most of the elaborations mention they don't think it constituted a full story - that it read more like the first chapter of a book.
To which I usually respond (silently): "Sometimes that's what a short story is - the first chapter to a book that hasn't been written."
I find it wonderful that so many people have expressed desire for an extension to that story. I'll gladly write it (I aim to please. If there's an audience for a story, I'm going to give it to them). However, I disagree that it needs it in order to be complete. The idea has been enunciated. The picture has been formed. As a short, I don't think it requires a complete character or story arc in order to stand alone.
This has come up a lot with me lately as I write additional short stories for a planned collection. Some of them are complete stories in the traditional sense, but many of them aren't. I keep wondering whether or not to address this issue within the collection itself through some kind of authors note. I suppose it must be a consequence of being an indie publisher that makes me want to prepare the reader's expectations so much...