A double-spend is just what it says.  When you manage to spend the same coin twice.  I send a coin to amagi metals for a bar of gold.  Send the same coin to agora commodities for a bar of gold.  They both send me the gold.  Then amagi doesn't get their coins because of the double spend.  If amagi catches the double spend attempt and cancels the order before shipping gold, then it wasn't a successful double-spend.  It hasn't been spent twice, where one spend was fraudulent.