First of all, welcome to the forum. As I am sure you have seen many times, take a moment and breathe. You have time and options for dealing with this.
I don't believe your circumstance is special. I am fairly certain I have read at least two or three other stories like yours here.
Even though web developers will often consider a file non-public if it is not linked from anywhere in html, there are still a few methods that it can be publicly revealed. If you are able to open a browser, type in the URL they sent you in the screen shot, and be taken to an image without any type of limitation, then technically it is publicly available. There is nothing stopping it from happening like password protection, encryption, etc. Directories on a website can be, and often are, indexed even when there is nothing pointing to the contents within.
For example, anything in the root directory of your site probably will not be indexed because you have an index.html, index.htm, index.php, default.html, etc. However, let's say that you have a header graphic in a folder /images, that is something like /images/header.jpg. You link to this and a robot scanning the site can see that you have linked to this. So it removes the header.jpg and just visits /images because it now knows this directory exists and contains at least one thing. At this point, if you have not instructed your web server to display an error for a directory that does not have a default when something visits it, your web server will often list out the contents of the directory instead - including sub folders and files. See
http://httpd.apache.org/docs/current/mod/mod_dir.html for a brief description of how the Apache web server handles things. You have likely run across these listings in your own browsing experiences in the past. The links to the files and directories are clickable and work just like a normal website. So even though you didn't link to the file anywhere in your html, your webserver probably still told the Getty robot about the image in one way or another.
Your story is similar to others. Keep reading to see how they have handled theirs. A few discussions that may be relevant:
http://www.extortionletterinfo.com/forum/getty-images-letter-forum/possible-reasons-for-class-action-suit-against-getty-in-the-usa/40/http://www.extortionletterinfo.com/forum/mccormack-letter-forum/never-received-getty-letter-but-got-mccormack-email-help!/http://www.extortionletterinfo.com/forum/getty-images-letter-forum/received-letters-from-ceg/10/