Yes, they are. They are generally expressions of an individual's intent. A health care proxy is a more comprehensive document that names an agent to make health care decisions for the principal. The agent has broad powers to act for the principal in the event of incapacity.
A living will is not legally binding in New York. A much better choice would be a health care proxy which is legally binding.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on since each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.