Initially, the midwives/doctors will just take your date from the first day of your LMP, so, if that was 17th April, then you'd be 5+3 today (which makes sense for the strong BFP). The LMP calculations are based on an average 28 day cycle with implantation occurring on day 14 so it's not 100% accurate but it gives you a guide.
Any spotting or bleeding after that may just be implantation bleeding which is not unusual and usually happens in the days/week after ovulation rather than on the day of ovulation itself.
You won't get an exact EDD until your dating scan as that goes by the actual measurements of your baby and normally ties in more with when you ovulated. For example, I know I ovulated around day 19/20 of my cycle and sure enough, the dating scan put me 5 days behind the date LMP gave me.
Don't forget, whenever you ovulated, you add 2 weeks to the beginning as it's taken from when the egg is released. That means that when you actually conceive, you are already 2 weeks pregnant! Also, sperm can live inside of you for a few days (up to a week I believe) so if you DTD on 4th, you could have ovulated later.
Hope that all makes sense.