logical deduction - When is Cheryl's Birthday?

I saw this question on Facebook:

Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates.

May 15 16 19

June 17 18

July 14 16

August 14 15 17

Cheryl then tells Albert and Bernard separately the month and the day of her birthday, respectively.

Albert: I don't know when Cheryl's birthday is, but I know that Bernard does not know, too.

Bernard: At first, I didn't know when Cheryl's birthday is, but I know now.

Albert: Then I also know when Cheryl's birthday is.

When is Cheryl's birthday?

There is an official solution which is explained in the link below, but I don't understand the logic.

July 16, as explained here. https://www.facebook.com/kennethjianwenz/posts/386479228197631

What is wrong with the logic in the answer I posted below (I put it in spoilers if others want to try)?

Answer

I will try to help you understand the answer. It is important to note that Albert and Bernard both know the answer before we do.

With Albert's first statement, it is clear to us that Albert was told either July or August. Had he been told May or June, he would not be able to state definitively that he knew Bernard didn't know. (May 19 and June 18 both could be uniquely identified immediately by Bernard without Albert's help, so in order for Albert to know that Bernard doesn't know, the month he was told must not be May or June.)

Because Bernard has been able to identify Cheryl's birthday after Albert's statement, he must not have been told 14. Because Albert's statement revealed July and August, had Bernard been told 14, he would still be unclear on the date. (If he said he didn't know, then Albert would know the birthdate, but Bernard would never be able to deduce it.)

We still don't know whether it is July 16, August 15, or August 17.

However, since Albert (who was only told the month) is able to state that he also knows the birthday, he must not have been told August. Had he been told August, he would be unable to decide if Bernard was told the 15th and 17th.

It is only after the third statement that we (as outsiders) can deduce the birthday. Bernard knew after the first statement and Albert knew after the second.