hey hey, I know you!
So glad you have recovered so fast...what a miracle (and stubborn genes I guess).
Believe me I understand the feeling of your family not understanding the whole bike thing. The difference is that I was pretty out front with my family from the get-go about my bicycle commuting. They are still uneasy about it but have come to accept my lifestyle choice. I have thankfully not been in an accident in my three + years of riding, so I don't have that situation to contend with.
I would tend to think that most of the reason that they are irrational about your biking on city streets is because of ignorance. I believe this also stems from the fact that they do care for you, but like all families, tend to be overprotective and irrational at times. Also, that feeling was probably intensified since you were hiding your biking from them.
So two things:
1. You need to start believing that you can love and respect your family and also be an adult and disagree with their opinions. Be forthright with them and share your positive reasons for cycling (health, environment, changing the street culture by being a presence!). Don't be afraid to be who you are or that will show and this alone will cause your folks to be more wary of your choices.
2. Get informed with the facts of bike versus car safety...and your right to be on the road... and you'll be able to share this information with your family, showing that cycling isn't quite the horribly dangerous and immature thing to do. Emphasize that it wasn't your riding a bike that caused your accident, it was the irresponsible idiot that ran you down... and that could have happened with you in a car as well.
This isn't easy, but you can do it! And you aren't alone with that sort of situation. We should ride together sometime and work out some of those nerves if your body is up to it. 
~Tricia