Pregnancy is not counted by your doctor from the day you had sex, it's counted from the first day of the last period. Not because the doctor thinks you were pregnant on the first day of your last period, he knows you weren't pregnant then, you were having a period! but that is where the count begins.
If your doctor said your due date is May 13, that suggests conception was August 20, which lines up with your last period being August 6, which is when you say it was. You said your had your first ultrasound "at 6 weeks." 6 weeks starting from what?
Sorry not 8 weeks 9 days 8 weeks 6 days