The reason for different behavior is that, IE uses ActiveX version of Flash Player where other browses like FireFox, Google Chrome, Opera etc uses Plug-in version, that's why you are getting Flash Player is installed in FF. However from the screenshot that you attached for Internet Explorer (IE) browser seems that you are running IE with disabled Add-On (Please refer to the attached screenshot) that's why you are getting Flash Player not installed, so please try opening that site in IE which does not have disabled Add-On and check?