Mr Phoenix, your source was correct. "How do you do" is a greeting used only when meeting someone for the first time. It is not so rare. In fact, I use it often when meeting people for the first time. Though it is a question literally, it does not function as one. The phrase is simply an established convention. The smoothest response in every case would be, "Nice to meet you."
Mr. ___: "How do you do, Mr. Phoenix."
Mr. Phoenix: "Nice to meet you, Mr. ___"
"How do you do" is not the same as "How are you." Mr. Maxey's answer is absolutely correct for "How are you," but not for "How do you do."