Carter names the person who hauled goods by cart or wagon—Anglo-Norman careter from a Latin root for cart. Before trucks and trains, that haulage job was ordinary infrastructure, which is why Carter became a frequent English surname across Britain and Ireland.
It is occupation, not "son of Cart." Cartwright ("cart maker") is a related craft name but a different surname. American Carters inherit the English job-word in the great majority of cases. Families carried the name across the Atlantic without changing its core sense.