Ah I see what you mean now. I found quite a few on paper. This could be an interesting computer programming exercise, but I'm a bit busy at the moment and wouldn't be able to write one for maybe a week. We know the area of each congruent part must be 4. There are only so many possibilities.
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.
Suppose the board is represented by lattice points on . For counting purposes we would need to consider whether two boards are equivalent up to rotation/reflection. I made an image that lists rotationally/reflectionally equivalent boards separately (no time to make it look pretty): (Click on image for full size.)
Don't know how many are possible, just got this from playing around a bit.
Edit: In the first row, the last two boards can be reflected to obtain two more boards.
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.
Yes I addressed this in my post, in less words than you did, lol.
That's an interesting story. As for formulas, sometimes you get pretty formulas, other times it's more of an algorithmic solution that may not look as nice.
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?