Your question is very important to discuss with your pregnancy care provider (midwife or doctor) because having an accurate due date affects pregnancy care decisions. Your providers will ask you more questions that may be helpful to provide a better idea of what was happening early in your pregnancy.   Generally, there are two possibilities based on the information you describe.  The bleeding you had June 2-6 could have either been a period, which came earlier than you expected, and you weren’t pregnant or you may have been pregnant and had early pregnancy bleeding. 
As many as one in four women have spotting or bleeding in early pregnancy.  This bleeding is not the same as a period but has a pregnancy-related cause.  A very common and normal reason for bleeding which occurs 10 to 14 days after conception is what is called implantation bleeding. Implantation is when the fertilized egg implants in the lining of the uterus.  Based on the dates you mention, the bleeding you had on June 2-6 could have been implantation bleeding if you actually conceived on May 17.  The very best way to know how far along a woman is in pregnancy, when there is any question about last period dates, is with an ultrasound. Ultrasound is most accurate to predict a due date when done in the early stages of pregnancy, optimally before 22 weeks (since the first day of the last period).  When a woman’s first ultrasound is done after 22 weeks gestation the information from the ultrasound about the due date may be in error by two weeks or more.