I see "No file chosen" although the image is uploaded to Cloudinary. Why is that?

While the upload works successfully, once the upload has been initiated, the default behavior of the upload component is indeed to reset the file input so it looks like nothing happened.
This behavior can be changed by calling $('.cloudinary-fileupload').cloudinary_fileupload({replaceFileInput: false});.

You can bind to cloudinarydone to decide what to do when the upload is complete, e.g. show a thumbnail.
This can be also used for programming UI changes, you can also customize the look & feel and the behavior of the upload flow.

Please have a look how this is done in our sample project (for example, in Python):
https://github.com/cloudinary/cloudinary-django-sample

Also, you might want to check out our out-of-the-box upload widget:
http://cloudinary.com/documentation/upload_widget