Hello and thanks for your Social Security questionCustomer
There are specific dates of enrollment for Part B of Medicare. If you miss the general enrollment at Age 65, meaning you sign up more than 3 months after your 65th birthday, then by law, the coverage starts the NEXT July 1.
Your application must be in no later than MARCH 31 of the current year to be eligible for the current JULY, so effectively any application received between APRIL 1, 2013 and MARCH 31, 2014 would not start Part B coverage until July 1, 2014.
Creating an online account in the SSA.GOV website is not enough to enroll in the Medicare programs. You have to separately enroll.
Based on the fact that you enrolled on 5/13/2013, the start date of JULY 1, 2014, is correct and cannot be changed.
The publication at the link below provides verification of this information on Page 12.
I am sorry that I did not have better news, but the Medicare rules in this regard are pretty clear,