With respect to the term “Supreme Court” as the name of the trial court, it dates back to reforms to the English model of courts, inherited by the colonies.
In England, there were three royal courts of common law, termed the superior courts: King’s Bench, Common Pleas, and Exchequer. In theory, King’s Bench was public law and criminal cases; Common Pleas was private law suits, between subjects, and Exchequer was the royal revenue court.
There was a fourth superior court, the Court of Chancery, which adminstered equity, not the common law.
The usual pattern in English colonies was to copy the mother country’s court system, so most colonies had a King’s Bench, Common Pleas, Exchequer, and Chancery court. New York followed this pattern, as the link that Bibliophage provided shows. However, in a small colony, with a limited number of qualified lawyers, having so many courts served little purpose. Eventually, the colonies tended to unify at least the three common law courts into one court, and sooner or later added in the Court of Chancery.
The unfied superior court was commonly called the Supreme Court, not because of its position in the court hierarchy, but because it had general original jurisdiction - any claim or prosecution could be brought in the Supreme Court, unless the matter was assigned to some other court by statute. So, it was supreme in that sense.
The unified superior court was supreme in another sense. At common law, appeals are entirely creatures of statute. Unless a statute provides for a right of appellate review, the decision of the unified Supreme Court, as a superior court, is final. (At common law, decisions of inferior courts could always be reviewed in the superior courts, so those courts weren’t “supreme” in this sense.)
The development of the New York Supreme Court in the link is paralleled in other former British colonies, and in England itself. England unified the superior courts in the 1870’s, and called the unified court “The Supreme Court of Judicature.” It is not the final appellate court for England - the House of Lords, sitting in its judicial capacity, is the ultimate court of appeal for England.
In Australia, the superior court of original jurisdiciton in most of the states is called the Supreme Court. The ultimate appellate court is the High Court.
In Canada, the nomenclature is a bit mixed. In Newfoundland, Nova Scotia, Prince Edward Island, British Columbia, Yukon, Northwest Territories and Nunavut, the superior court of original jurisdiction is called the Supreme Court.
In Ontario and Quebec, it’s called the Superior Court.
In Alberta, Saskatchewan, Manitoba and New Brunswick, it’s called the Queen’s Bench.
The ultimate appellate court is the Supreme Court of Canada.