To close this (even if a little late): borger where the people who lived in and around a borg, and borgmester seemed to have been either the master/leader of the castle-dwellers, or the lord of the castle himself.
The same counts for the German words Burg, Bürger, and Bürgermeister.
In Italian the word to define a district is "Borgo" , it was used in older time to refer to villages.
It comes from the latin "burgus", I guess that the German word comes from here too.
The only difference is that in Italian it's not used to refer to citizenship, except for the word "in Borghese" (wearing plain clothes) or "Borghesia" (the middle class of society).