In general all the stories I've written (the two or three of them :p) start with the same context: there's a single, alone character, who is placed at a setting or circumstance that they know well but that is expanding into something unknown or unannounced. The setting or circumstance could be something discrete and defined, like a room in a hotel or a road to a town, or something abstract and fuzzy, like the beginning of a raid; but the character is always conceptually alone and that allows me to describe the expanding "setting" in terms of their perception of the changes.
I'd say it's... not exactly a good way to work with short stories though, as it is kinda hard to set up and only works well when the beginning is a In Media Res kind of thing. If I had to start short (very short) stories in a particular way, I'd probably choose something like the finishing of an exchange / interaction that justifies or explains the plot of the story (Star Wars IV shows more or less how it's done).
My stories always tend to start with a character on their own. Even when they're in a group of people/creatures or with someone else, they're always alone for some reason. I tend to write about loners a lot, apparently!
I also always start my stories as close to the first cause of action as I can get. Whether it's stumbling across a destroyed village, heading off into a Pokemon battle, or sneaking off through dark corridors, the story's going to start soon.
Short stories I can't really explain how and when I decide to start them. I just know how the story's going to start.
I start off normally with one character. It's usually nothing exciting, just the character getting ready for the day or something. Then they exit their house and you get to know where they live. Of course, this is my strategy for chaptered fics and longer stories.
As far as short stories go, I don't have much experience with them. From what little work I have done concerning them, I usually start off with the story right away. Such as an ambulance driving to the scene or something like that.