He divorced her in front of her family then in the presence of the notary public. Does that count as two divorces (talaaq)?

Firstly: 

Divorce at the time of anger may or may not count as such, depending on the type and intensity of the anger. For a detailed discussion of that, please see the answer to question no. 22034. 

With regard to the divorce that happened before the notary public, if the husband intended thereby to confirm the divorce that happened in front of your family, or he wanted to give notice of that divorce, then it is not counted as a second divorce. 

But if he wanted to issue a second divorce that was different than the first, then it is as he intended. What counts in this case is the intention and aim of the husband. 

The divorce that happened in front of the notary public – if the husband intended it as a new divorce – is regarded as a divorce if it happened within the ‘iddah. The majority of scholars are of the view that it counts as such, because the woman who is observing ‘iddah of a revocable divorce comes under the same ruling as other wives so divorce is counted in her case. 

It says in al-Mawsoo’ah al-Fiqhiyyah (29/12): The majority of fuqaha’ – Hanafis, Maalikis and Shaafa’is, and it is the view of the Hanbalis – are of the view that divorce is counted as such with regard to a woman who is observing ‘iddah of a revocable divorce. If a man says to his wife with whom he has consummated the marriage, “You are divorced,” then he said during her ‘iddah, “You are divorced,” a second time, then that is two divorces (talaaq), so long as he did not intend to confirm the first divorce. But if he intended to confirm the first divorce, it does not count as a second divorce. End quote. 

Thirdly: 

If the divorce mentioned happened when the woman was menstruating or during a period of purity when her husband had had intercourse with her, then there are some scholars who stated that this does not count as a divorce. See question no. 72417. 

Fourthly: 

If three divorces have taken place, and the woman has become irrevocably divorced from her husband, then she is not permissible for him unless she marries another a husband in a real marriage, not a marriage aimed at circumventing this ruling, and he has intercourse with her, then dies or divorces her. In that case the first husband can marry her in a new marriage contract and with a new mahr. 

But a marriage aimed at circumventing this ruling (known as nikaah al-tahleel) is invalid and is haraam, because the man is not marrying her because he wants to marry her, rather it is in order to make her permissible for the first husband. The Prophet (peace and blessings of Allaah be upon him) cursed the one who does that, or for whom it is done. 

A tahleel marriage does not make the woman permissible for her first husband. 

Ibn Qudaamah (may Allaah have mercy on him) said in al-Mughni (7/137): A marriage aimed at making a woman permissible for her first husband is haraam and invalid, according to the majority of scholars, including al-Hasan, al-Nakha’i, Qataadah, Maalik, al-Layth, al-Thawri, Ibn al-Mubaarak and al-Shaafa’i. 

He also said: A marriage aimed at making a woman permissible for her first husband is invalid and it does not make her permissible for the first husband. End quote from al-Mughni (7/140). 

And Allaah knows best.