I cut the 5x5 square into 4 smaller squares and going from the center dot that is common to the 4 small squares and then numbered each dot and tried different number combinations. This helped me get congruent pieces except for the the one that doesn't start from the "center" dot- the one that divides the 5x5 into 4 pieces that are each 1x4.
I love "playing" with math. You listed as in row 2 figures 1 & 2 rotations which you counted as different designs based on construction which are simply rotations of each other but are disarded as dupicates of the same piece since they are not uniquely different pieces. This reminds me of a problem from the Gateways pre-algebra text that asked of hexamino shapeswhich shape has the smallest perimeter and which the largest. The answers were intuitive from prior experience but I was surprised to find that there was more than one shape that yielded the same perimeter. I was then determined to examine all variations to show that there were no other surprises lurking in the problem and I was at a math conference where one of the authors of the book spoke and I asked him if there was a formula for how many ways you could uniquely arrange the 6 squares edge to edge. When he said that there was, I asked him if he knew what it was off-hand he said that no one has found it yet but that since it was a natural formation it had to have one.
Well like I said I have other things to keep me busy but this could be an interesting programming exercise to find all different configurations.
So I understand that you prefer to consider two configurations that differ only by rotation as duplicates. How about reflection?