Dear TM,
 
Point-based triangle reduction simply removes points (and the corresponding triangles) and retriangulates the holes created. It will do so with points which alter the geometry only within the preset tolerance. (Usually this means that small triangles will disappear or triangles in flat areas.)
 
Edge-based triangle reduction will remove edges by collapsing the edge to one point. Each collapse will in total reduce the number of triangles by two and replace two points by one. Again the edges that have only a limited effect on the geometry are handled.
 
The advanced edge-based method does basically the same as the “primitive” edge-based method: It collapses edges. It will however apply a more advanced method to calculate the new point. It uses some higher order fitting to determine the position of the new point (that replaces the two edge points). Furthermore, it keeps track of possible systematic changes to the surface over the different iterations and corrects for that afterwards.