Back in the day, my professor Helio Gracie agreed to begin the first jiu jitsu federation in Brazil. The starting of that federation allowed for the growing of sport jiu-jitsu and soon became the "brazilian jiu jitsu" that we know today.
Gracie Jiu Jitsu is the martial art that the masters Carlos and Helio Gracie brought into existence. Through the sweat of their brows they revived techniques from Japan and created a fighting system unlike anything previously seen in the world.
I recently watched a movie called Ip Man (Chinese martial arts master). While the fight scenery was obviously staged, the intentions of Gracie Jiu Jitsu were to empower normal everyday people to be able to actually fight and defend themselves much in the same way Ip Man (in the movie) would use his Wing Chun to win duels and defend himself from thugs. (defeating multiple opponents, however, is a topic that I am not touching..)
Gracie Jiu Jitsu is a martial art meant for fighting much like any combative system like Kali, Silat, Kyokushin..(the list goes on).
Brazilian Jiu Jitsu (if we must use another name) is the sportive version, the recreational side of our art that everyone can enjoy without going out and picking a fight or going into MMA.
Sport Jiu Jitsu, however, was the next step in Jiu Jitsu's proliferation and popularization. Now we have people contesting which techniques will work on the street and which can only work on the mat. These arguments must be discussed with a universal understanding of what we are doing here..
Positions such as the deep half guard, 50/50, x-guard and SO on are all evolutions of the art of grappling. They were quick solutions to certain problems in basic grappling and the playing of these guards and positions are only possible because the rules and limitations of our sport allow for them. This much is simple. When you take out the punches and the nhb, these positions are naturally there. I believe when it comes to bettering ourselves at grapplers in sport, there is nothing we should not learn. I am 52 and I still open my mind to any white belt-black belt that has a new move to show me. But when it comes to using those techniques for the actual act of self-defense and fighting.... I am sure many players and fighters would quickly agree that they would not use their favorite half guard when knocked to the ground in a fight. It is just common sense for anyone who values their face :-)
In conclusion, it is simply understanding what you are training that makes the difference. There is no harm in training for self-defense or sport as long as you understand the difference. Like many things, jiu jitsu does not have to be a chore.. it can simply be enjoyed.