The sophisticated way of saying your name is "The name is..." You simply do not say "My name is..." There is nothing wrong with the latter but it is too casual and less subtle. As a matter of fact, 99.99% of the time you should hear people say "The name is" or "Name is...". And 100% of the times when Bond introduced himself, he said "The name's Bond, James Bond."
Minor deviation makes big difference.