What is the relationship of 'added edges' to existing 'faces/edges'?

example:
I made a block with 10 divisions on the X-axis and 10 on the Y-axis.
Then I wanted to Cut Out a diamond shape from one of the divisions so first I tried by adding the edges of the diamond, then 'filled holes' between the new edges.
BUT when I tried to then Cut the diamond shape out I found all I could do was cut either the whole 'square' or just the 'faces' surrounding the diamond leaving the diamond that I wanted removed.

OK, I finally figured out I was working in the 'wrong direction' (from inside to outside) and finally just deleted the whole 'square' division and then added Edges to form my diamond and then 'fill holes' to add back the outter faces. Done deal, I got what I wanted....

But now I am not sure just what is the relationship of "add edge" to the existing mesh if the added edges to not actually divide the face they are formed across??
(hope this makes sense?)