I believe the name "short circuit" comes from the notion that the electricity is taking an undesired "shorter" path rather than going through the load.
Note that this may not actually be a physically shorter path, just an undesired alternate path of low enough impedance that a significant portion of the electricity flowing is bypassing the load.
As for the terminology in the article (North America power described, may differ in your area):
Background information (simplified) -- Main electricity in NA is delivered in three main forms:
Double phase - Two 120v AC signals 180 degrees out of phase between a ground-potential NEUTRAL wire and two HOT wires.
This is how power is usually delivered to residential and non-industrial business.
This can be used as a single 240v source, or as two ...
Single phase - 120v AC between NEUTRAL and one HOT wire.
This is almost always 1/2 of a Double Phase feed.
Three phase - Three phases 120 degrees out of phase with each other (voltage depends on service)
This is how power is usually delivered to large industrial business.
Back to your article, my interpretation:
"Phase to ground" means any HOT wire in a feed unintentionally connecting to ground.
"Double Phase" to ground means both of the HOT wires in a two-phase feed unintentionally connecting to ground.
"Three phase to ground" means multiple HOT wires in a three-phase feed unintentionally connecting to ground.
"phase to phase and three phase" means two or three HOT wires in a feed unintentionally connecting to each other.
Examples where multiple phases could short at the same time (or close in time) are things like fire where insulation is burned off, submerged cables where the shielding gets compromised and all wires deteriorate together, concurrent physical damage (backhoe cuts the cable) etc.
Dave