I like your logic (I think), but I keep arriving at July 16.
I’m a little doubtful about how Bernard makes his claim, but Alert’s claim confirms the answer.
Albert and Bernard each have been given correct data.
Albert was given the month but, as there are multiple possibilities within each month, he can not determine the date.
Bernard was given the day. Possible days are: 14, 15, 16, 17, 18, and 19. Except for 18 and 19 there are multiple occurrances of each day.
Albert announces that he knows that Bernard can not know the date. This tells Bernard that the day is not unique. Therefore May 19 and June 18 can be ruled out.
Bernard said “at first I didn’t know, but now I do”.
At first his information, the day, was not unique.
At first Albert’s information, the month, was not unique.
But now after Albert’s announcement and consequent ruling out of May 19 and June 18 Bernard claims to know the answer. Then Albert, who knows only the month, says he too knows. The only month left with a single day is June. Albert’s claim can only be supported if June 17 is the answer.
That’s my story and I’m sticking to it. ;)