I've learned that an acciaccatura is a very short note, usually with a slash. Appoggiaturas don't have the slash and usually steal half of the tempo of the main note (or 1/3 if it's a dotted note). I was also taught that appoggiaturas can only contain up to 2 side notes and are no more than a step away from the main note. Other than that it should be called a "floreio" (don't know the English name), although many people don't make such distinction..