Thank you for the feedback! It turns out that I did a couple of things wrong.
First, I completed a fresh install of WordPress on a domain with an HTTPS prefix rather than on the more typical HTTP. The URL for every post and page on the blog started with HTTPS. It's worth mentioning that I purchased a valid SSL certificate.
I restored post data from my old WordPress install and found that the embedded images weren't loading because the URLs started with HTTP rather than HTTPS in the html syntax. My solution was to simply revert back to HTTP in the WordPress address and the site address rather than manually edit each post.
Afterwards, I installed the WordPress HTTPS plugin and enabled SSL in the administrative backend and in certain pages where the user enters sensitive information (such as a username and password).
Little did I know that the htaccess directives were incorrect all along...
RewriteCond %{HTTP_REFERER} !^http://(www\.)?somewebsite\.com/ [NC]
Notice that the URL only starts with HTTP, unlike the URL in your example where the rewrite rule is able to accept both HTTP and HTTPS. I made the change and all the thumbnails appeared as they should in the "Manage Galleries" control panel for FlaGallery.
However, I now get a blank page whenever I attempt to edit a thumbnail in FlaGallery.
I've discovered the following error, but I haven't found a solution to rectify it...
XMLHttpRequest cannot load http://example.com/wp-content/plugins/flash-album-gallery/admin/manage_thumbnail.php?id=1&random=1341879773544. Origin https://example.com is not allowed by Access-Control-Allow-Origin.
One solution is to simply disable SSL in the administrative backend and in the dedicated login page, but I'd rather have it enabled.
I tried tweaking FlaGallery by adding two lines of code I found in another post regarding a different plugin...
header('Access-Control-Allow-Origin: '.get_site_url());
header('Access-Control-Allow-Origin: '.str_replace('http://', 'https://', get_site_url()));
I've had no luck! Any suggestions?