OK, I just checked on Firefox and it seems to happen there as well. With the "Show URs/map problems (LMUR v0.6)" initially disabled, or if enabled, it does not seem to make any difference. I have also checked when going to Livemap from the editor, just to ensure I am logged in already.
The script itselfs works fine, shows the overlay if enabled & tells me if I'm not logged in if I attempt to initially enable it.
I'm on the world server, and cannot see any errors in the console.
I'm just wondering if it's subtle differences again based on where in the world you are located. Can you remember the 'Livemap' URL in the editor for me in South Africa that just pointed to Livemap
& not the same location as that you were editing ? That's why you added the Livemap link in UROverview.