Thank you Joseph for that great riddle. It took me some time to work it out, but I think I have it.
The correct date is 16 July.
Here is my thought process:
It couldn’t have been May, as then there would have been the possibility of Baruch knowing the birthday, namely if Chaim would have said 19, which is a date which only appeared in May.
For the same reason it couldn’t have been June, as then there would have been the possibility that Baruch was told 18, which only appeared in the June list.
This is as far as Avrumi could have known at this stage.
However by him saying so, Baruch now knew that May & June were out. But it could still have been July or August.
Now, if he would have been told 14, he still couldn’t know the correct month, as that appears in July & August. Therefore if he was told 16, he knew it was 16 July; if he was told 15 or 17 he knew it must be August. So he now knew the correct date.
Because of Baruch’s statement, Avrumi now also knew that 14 July was out.
If Avrumi would have been told “August”, he still wouldn’t have known which day in August. From the fact that he claimed to know, he must have been told “July” and the birthday must have been 16 July. Mazzel Tov!