I just figured out that the popup Login window does not display on my homepage, for all browsers, whenever it loads as an HTTP page *when* my site is setup for HTTPS. Ever since I added code to .htaccess to automatically convert HTTP to HTTPS the homepage popup Login window has worked correctly.
To .htaccess I added... under RewriteEngine On
RewriteCond %{SERVER_PORT} 80
RewriteRule ^(.*)$ https://my-site-name.com/$1 [R,L]
