For me the difference between "street" and "sport" comes down to two basic things: the goal and the time.
The goal or object of "sport" is to out point, submit, choke out or KO your opponent. The goal of "street" (with regard to self defense) is to go to work tomorrow. I just need enough fighting to get to the point where I can go to work tomorrow. Maybe that is deescalating, maybe that is giving up the cash in my wallet, maybe that is running away or the worst possible thing... apologizing and backing down. In "sport" I am working towards points and submissions, putting myself back into the fight where on the street that is looking for the out. The other guy hits me, knocks me down, but I can get up and out the door.... I don't have to re engage with him, as I would in "sport."
Yes, it gets more complicated when your family is around. Then your goal is still not to defeat the other guy, but get your family to safety. You may be better off stalling him, making him focus on you, while your family gets to safety, then you get yourself to safety.
The deal is that when training for self defense / "street" the idea is to get to the point where you can leave... and then leave at that point.... don't hang around to "defeat" the other guy.
The second thing is the time. In a sport, many times you start the competition with a feel out round... where you are trying to figure out what his tendencies are, how much power he has, how fast he is.... You have time to set him up, by showing tendencies, baiting him in.... In the "street" you need to hit your highest percentage move for that situation, as soon as you can, and then get out before his friends show up, before his weapons come out, before he gets the upper hand. You don't have time for the "feel out round."
Another issue with time.... in "sport," you both know when it will start.... you are usually asked if you are ready. On the "street" the confrontation has already begun, before you know it has. They have picked you out as the victim, decided on their tactics.... all before you realize that your are in the middle of something. Being situationally aware helps you to recognize sooner, when you are getting involved in an altercation. So, in "sport" you both start off as equal.... in "street" the other guy has done as much as he can to start off in the advantage. So, you need to work from the position of disadvantage to the point where you can get out.