It is quite normal to not have your period at 13 (I do not know why your doctor told you otherwise).
If you started developing breasts, your period will still come. Sometimes girls are also ashamed to admit they are not yet having their period and they will tell you they have - so don't worry about the other girls.
Periods are also very irregular in the early stages and a girl can have her first period and then nothing for months. So a number of your friends can tell you they have their periods but only had it once or twice.
Your friend probably started spotting. It is bleeding that is very light and you can normally only see it when you wash or on toilet paper when you wipe. Spotting is a normal part of life for a woman.
If the white stuff is looking like cottage cheese and makes you itch and irritated, you might have an yeast infection. It can be caused by many things like wiping from the back to the front on the toilet, diabetes, obesity and may others.
If it is just white fluid and it smells fishy, it is a bacterial infection and you need to get antibiotics from your doctor.
Most likely it is just your hormones beginning to work. Right through your life you will experience different secretions from your cervix. This indicates that your hormones are working and it causes your ovulation and menstruation at the right time.
I am sure your mom can help you identify it if you have an infection.
Bets of luck!!