New York State doesn’t have third degree murder, but it does have second degree murder which includes the following:
https://statelaws.findlaw.com/new-york-law/new-york-second-degree-murder-laws.html
Pursuant to the statute, a person commits second-degree murder in one of five ways:
1: with the intent to cause the death of another person, he or she causes the death of such person or a third person;
2: under circumstances demonstrating a "depraved indifference to human life," the defendant "recklessly engages in conduct which creates a grave risk of death to another person, and thereby causes the death of another person";
3: acting alone or in concert with others, the defendant commits or attempts to commit a specified felony (including robbery, burglary, kidnapping, arson, rape, and sexual abuse) and, in the course of and in furtherance of such crime or of immediate flight therefrom, he or she causes the death of a non-participant;
4: under circumstances demonstrating a "depraved indifference to human life," a defendant 18 years old or more "recklessly engages in conduct which creates a grave risk of serious physical injury or death" to a person less than 11 years old and causes the death of such person;
5: or while in the course of committing a specified crime such as rape, a criminal sexual act or sexual abuse, a defendant 18 years old or more intentionally causes the death of a person less than 14 years old.
Depraved Indifference and Felony Murder
The second type of second-degree murder listed above - "depraved indifference" murder - is a form of reckless homicide.To convict a defendant for second-degree depraved indifference murder, the prosecution must prove - in addition to the defendant's creation of a "grave risk of death" to another person - two distinct mens reas, or mental states: recklessness, and a "depraved indifference to human life."
Convictions for depraved indifference murder have become increasingly rare, and often involve conduct creating danger to a group of people rather than a single individual.