there are VERY subtle differences
"I ran out of salt." I had to use salt for other purposes, and sorry, there's no more.
"i'm running short on salt" I still have some salt, but very little. So little, I can't spare it.
"I used up all my salt" I cooked something that used the remainder of the salt.
"i'm all out of salt" This is what you might typically hear at a restaurant, its polite way of saying they don't have anymore salt.