When an image is cropped and the changes are saved, the binary of the asset is properly updated. But all the instances of the image (except in the crop tool itself) in the UI continue to show the old image. As soon as the editor is closed, all instances should be updated to reflect the actual image of the asset.

Where is the problem?
Note that if you get the url (right click) on the preview image of the asset after the edit, and paste it into a new browser tab - you will see the edited image properly. Then if you click on the thumbnail or item in the list or tree, the preview will be updated to the proper image.

This indicates that it is the browser that has a cached instance of the old image.

Suggestion:
After editing the image, perform ajax requests to get the new images in the browser. Once that is successful, perform a refresh on the preview and list view so that the images on the page are refreshed with the new images.

Alternatively an easier way would just be to append a "cache buster" Query string on all the image requests in the admin ui such that all images are always reloaded. i.e. a random, date seeded number image.jpg?cb=r83904834903.
But this will negatively impact the performance of the admin interface.
The preview and list would still need to be refreshed.