Maher or Muller in 1860 census

I am looking for info on Herman Muller (Armand in French), listed as father of Alfred Maurice Muller on his death certificate. Alfred was born about 1845 in France and died 1906 in NJ. His mother was Pauline. She lived in Paris where she attended school and met her husband. Family stories say that Maurice had relatives named Marie, Louise, and Calixte Mullaire who came to NY from Marseille.

I didnt find this family in any US CENSUS prior to the 1870 NJ census when Alfred is already married. However, he must have lived in NY because that is where he signed up for the war.
I found a very close match in the 1860 NY census:
Erman Maher married to Pauline. Kids Mary, Klist, Louisa, and Maurice.

I am wondering about the name Maher. Could this be spelled wrong on the census form, or was the name later changed to Muller? I am hoping to make a positive ID with this family, as it is the only lead I have!