I took a look at the psd file you provided and I encountered the same problems you did when trying to resize the image to the dimensions you wanted. I am uploading a new psd file of one that I recreated. There is the original group you had and then I added a 2nd group of what I added. The size of the image is set to 200x200 and with the group I added, I didn't notice any of the corruptions you encountered.
My thoughts: I don't know exactly why your circles were messing up when you tried to resize them. From what I could tell, it looked like you rasterized the layers and I think that is what is causing the problems. What I did differently was I didn't rasterize any of the layers. I left the smaller circles alone, but I duplicated the bigger circle layer and on the copy layer I converted it to a smart object, that way I could use my pen tool to draw a selection around the top portion. Once I had the top portion selected, I then used cmd+j (for mac) or ctrl+j (for pc). That cut the selection from the bigger circle and added the top portion on a new layer. Then I just added your layer styles for the color. Once I did that, I then resized the image with out the constrain portions option selected and resized it 200x200. Resizing it 200x202 seems very unnecessary because a circle should be symmetrical with its dimensions. I'm not sure if any of this makes sense, but I tried to explain it the best I could.
In summery, instead of rasterizing the circle, try converting them to smart objects. Then resize.