Having lived away from Bristol for so long, most of my conversations around footy are with fans of various clubs (Liverpool, Man Utd, Leeds Utd, Hull City, Burnley, Blackburn Rovers, to name a few...). I find there really is a sliding scale of knowledge, from those who have no regard whatsoever for any clubs outside of the Champions League spots to those who have in-depth insights into the Championship and below.
With the former, you'll get "How did Bristol do at the weekend?" (which does annoy me a little, as it just displays their ignorance of anything below the Prem) and I usually jokingly reply with "Which one? City or Bears?". They also tend to use simply "City" and "United" to refer to the Manchester teams.
With the latter, they might still refer to us as "Bristol" but will also often use "City" as well, as they know I use that term.
At the end of the day, non West Country football fans will continue calling us "Bristol" long into the future, so we'd better get used to it!