The right to trial by jury in criminal cases by a unanimous jury of twelve under the 6th Amendment to the United States Constitution and some related rights are only partially incorporated to apply against states via the 14th Amendment due process clause. (The right to a jury trial in a civil case under the 7th Amendment to the United States Constitution does not apply to the states.)
As noted, the federal right to a jury trial in a criminal case applies only when more than six months of imprisonment are a possible sentence pursuant to Baldwin v. New York. (Proceeding to trial without a jury on a more serious charge is allowed so long as the actual sentence does not exceed six months.) In practice, many states establish a right to a jury trial in many criminal cases where the U.S. Constitution does not require them.
Juries are not required in juvenile delinquency proceedings. The 6th Amendment right to a criminal jury trial does not apply to military justice either, although there are constitutional limitations that do apply to military justice. There are no jury trials in the territorial courts of American Samoa which is beyond the scope of 6th Amendment protection.
A unanimous jury of six or more jurors is allowed (at least in non-death penalty cases).
A state jury does not have to be unanimous (at least in non-death penalty cases). Ten of twelve is constitutionally sufficient for a twelve person jury; nine of twelve is not sufficient constitutionally. In practice, only one or two states permit non-unanimous jury verdicts.
There is not a federal constitutional right to waive a jury trial, although this exists in some states by state constitution or other forms of state law.
The right to have felony charges screened by a grand jury also does not apply to the states. About half of U.S. states require grand juries to screen felony charges, mostly in the eastern U.S.
I have the precedents in a criminal procedure text book and will update is I get a chance. See also the footnotes here.
To some extent, the constitutional provision in NY State is designed to make clear that bench trials are allowed when a jury has been waived. It also authorizes bench trials where the U.S. Constitution and state law permits them.
