Ian, I think it would help if you thought about cropping and sizing separately.
Cropping is the process of deciding how much of the original image you want to present. You might crop to remove wasted space, to remove distracting elements, to reposition the main subject(s) in the frame, to create symmetry, to fit a specified shape (eg an existing picture frame), or for other reasons.
Sizing is the process of deciding how many pixels you want to have in your image in order to suit the output. For example, for web display you might want to resize it so that the long edge is 960 pixels (Facebook standard) or 1024 pixels (many forums). My camera club sets a maximum of 1600 pixels for images to be projected. For a 12" print at 300 pixels per inch you need 3600 pixels, and so on. Sometimes there might also be a requirement to reduce the size of the file by compressing the data, which reduces quality.
I find that I often end up with multiple copies of an image: original full-size, print resolution, HDTV resolution for on-screen display, 960px for FB and forums, etc. That's OK.
Note that cropping is essentially an artistic decision and sizing is essentially a technical decision, and cropping logically comes first. That's why I suggest thinking of them separately. I know that some software allows you to do both in one step (eg crop to 8 inches square at 300 pixels per inch) but to my mind that's really not helpful.
The reason I mention this is that with your pigeon picture you initially solved a technical problem (how to make the image small enough to display) via an artistic process (cropping). And that created a totally different picture. Your two pigeon images are very different, even though they both come from the same raw data.
With that in mind I suggest you go back to the pigeon and think about how much of the original image you wish to present. How much of the bird's surroundings do you wish to show? Where do you want the bird in the frame? And so on. Then once you've decided that, you need to resize for display. (Personally I think the first one may be cropped too tightly and the second one has far too much wasted space, but these are subjective views and there is no right answer.)
Hope this helps.
PS Remember, always save the original. When you're trying out different crops, always do it to copies so that you can always go back and do something different. When resizing, always do it to copies so that you always have the full resolution available if necessary.
