The anger that ensued when the Cleveland Browns moved to Baltimore in 1996 still remains to this day. And there was outrage in the same manner when Baltimore was vacated by the Colts for Indianapolis in 1984.
The Browns left behind their brand in Cleveland, and became the Baltimore Ravens. The Colts, though, remained the Colts.
Expect the Chargers to remain the Chargers, despite talk that the franchise could "re-brand" itself to start anew in Los Angeles.