Unlike some other countries, US notaries (including State of New York) have no training in international documents. Also, according to this summary, New York State notaries are not allowed to certify copies at all. Even if you go to a neighboring state, you will probably find notaries will refuse to certify copies of birth certificates. Generally, even where notaries can certify copies, they are forbidden from certifying copies of vital records of their own state, and certifying foreign vital records is highly questionable.
Also, notaries cannot notarize a document which contains the notary's signature. So the translator must be a different person than the notary. The translator can swear in front of the notary that the letter and translation written by the translator are true, and the notary will prepare and sign a short statement that the translator appeared before the notary and took an oath.
I have never dealt with any immigration matters, but the instructions the original poster linked to seem to indicate that only a copy of the birth certificate is required. So that would make three birth-certifiate documents to be submitted: (1) a copy of the birth certifcate, in Chinese, (2) an original English translation, (3) an original letter from the translator certifying the translator is conversant or fluent in both languages, the translation is accurate, the other required details about the translator, and the notary's certificate that an oath was administered to the translator.