Interesting, Carol. I've often wondered this myself, shooting the DA*60-250 + HD DA 1.4x converter.
In your specific situation, I believe you're shooting JPEG straight out of the camera, right? One of the things I
think I notice with your non-TC cropped shot is that the sharpening effects are slightly amplified, as you would expect by enlarging a smaller area of the image to the same scale. That gives the
illusion of a sharper image than the uncropped TC image, but I wonder if it's really any sharper. I suspect the detail in the TC image might be a hair better, just not as visibly sharp.
I also think shooting
with the TC might have some benefits at higher ISO settings, as the higher pixel density for the same subject magnification should result in slightly less intrusive noise.
If you're shooting JPEG as I think you are, I'd be tempted to try increasing the in-camera sharpening on your TC shots, and see how those compare with the non-TC shots using your normal sharpening level. I think it's worth experimenting a little.
Aside from the above, all three shots look
great to me, so I don't think you could make a bad choice
Very impressed with the shots you're getting from that Sigma. Good lens + good technique = great results