When you select the 'mobile version' link on the main page, that takes you to http://humax/m/. mongoose looks in the directory first for a file called index.shtml which contains a redirect to /m/index.jim. That means the first time you see the mobile menu your browser is pointing at /m/index.jim. Lighttpd looks for index.jim first so the browser stays at /m/. When you click on the home icon, then you're on /m/index.jim.
It's just a configuration difference between the two but that's why you're seeing slightly different behaviour. You're right in that the underlying problem is a coding error. I'm upgrading to the latest version of jquery mobile and will rejig the code around to fix things before the next release.
Thanks for the explanation - nice to understand a bit of what is going on