Different browsers handle both Flash and SSL slightly differently, but if you have the HTML5-Only checkbox ticked in your settings, you can at least be sure the Flash handling is not the cross-browser inconsitency that you’re seeing.
The HTML5 app is not just a different UI, it’s also responsive and mobile compatible for all devices. The other UI runs on the Flash plugin, which is largely blocked by most browsers now that Chrome changed their default settings to block it in March (it can be turned on in settings but that’s not realistic for most users).
If the HTML5 app has everything you want, you may want to go with that only to be more consistent with all users.
Also, the html5 version is now SSL compatible as of yesterday!
Have you tried temporarily disabling all plugins except mystyle to test? That will tell you if it’s another plugin that is causing a conflict. By process of elimination you can find which plugin causes it and we could try to investigate it.