On the left is the final result I got by removing the 4 divisions where I wanted the diamond cut-out. I then drew in the new edges and filled the holes.

The left is a demo of my first (failed) method of trying to "add" the edges/fills and then "cut" the diamond out.

My question is:
On the Left image, what is the relationship of the Added Edges (in green) to the 'original' faces?
I had thought that the 'new edges' would have 'divided' the faces, but they appearently do not??

I'm trying to better understand the Point/Edge/Face relationship to get a better grasp on how to proceed when begining a project from scratch.

(also let me point out that I have no 'training' in modeling and just trying to 'blaze my own path' in my "spare time" ;-P