I personally compete in MMA, Boxing, and Grappling. I don't call any of them fights, and think it sounds weird. Fighting (at least between people) is what happens when people are too immature to control their emotions. The way I see it what I do is competition.
I use the word bout, compete, etc. I only use the word fight every now and then when I think it will help me better relate to a client, but even when referring to MMA it bothers me a little. My wife calls everything fights though. This doesn't bother me nearly enough to get an emotional response out of me
with that said you want to call MMA a fight but not BJJ it seems logical because one has striking, but really you're watering down your own definition if you do that. All combat sports have rules that limit what is allowed to be done. I can understand people that define fight as two opposing forces looking to defeat each other. In this case you can call any combat sports match a fight... to me it just sounds strange. more used to it from striking styles because boxing has been drilling that into the publics subconscious for generations, but it still sounds kind of strange to me.
To me people that consciously choose the word "fight" to describe their bout in a combat sport are doing so out of some subconscious desire to be viewed as a super tough badass. Which, in all honesty is probably the reason most people start training in the first place; for this reason as an instructor I don't call those people out. This is often helping them get past insecurities and achieve a status in their own mind that they never thought they could, being a "fighter" ... but at some point along the line that mentality should really change.