Dihedral group for arbitrary polygon Is a dihedral group only considered for shapes that when reflected or rotated fit exactly back into place of the original image? 

My confusion arises from this wikipedia article and specifically from this picture: 



Are these symmetries considered a dihedral group?
Those four actions form a dihedral group of order $4$. They are not symmetries of the letter "F" though, they are symmetries of a line segment (2-gon). The rotations are applied to the letter "F" in the picture to help you see what they do. If they are symmetries of anything in the picture, they are symmetries of the $y$-axis (or the $x$-axis).
Look, Michael, dihedral groups are easy, they are the next best thing after a cyclic group. While in a cyclic group you turn things around in a circle, see your dihedral group as a prism, like two identical cycles above each other, forming a kind of coin. Then your movements consist in turning the coin as if it were cyclic, and the extra moves being the flips of the coin and consider the "F" as a symbol stamped through the coin.