How much is a "huge" amount of current?
A better definition of a short circuit is a connection where the amount of current flowing isn't well understood by the modeling you are doing. For most things you are doing, 1000A is a "huge" amount of current. Staggering in fact. However, if you are running a conversion station that converts High Voltage DC power lines to AC, 1000A is actually quite small. They will have modeled the effects of that 1000A while you might not. Thus, for you, that may be a short, but for them it's standard operating procedures. Likewise, you may be comfortable with 500mA going through your circuits, but someone making a transistor a few nanometers wide might consider that amount of current to be a "short" because its going to destroy the transistor.
Consider a toaster. When you push the lever down, you create what would be considered a short: conductors from one side of the wallsocket plug to the other. And a whole ton of current does indeed go through it. However, toaster designers modeled this, and chose the length of conductor carefully, controlling the current draw, so that it doesn't flip the breaker. And, as you note, as it gets hot, its resistance goes up until it reaches a steady state.
Is that still a short? Depends on how comfortable you are with glowing yellow wires! The toaster designers clearly planned for it. And that's what really matters for a short. When you short things, the assumptions you made about how a circuit works go out the window, and you have to look at new novel limitations (like wires melting, sagging, and breaking).