It was a performance for the king in 1683, therefore private. However my reference book has this d being held to be the earliest existing English opera.
The 1656 performance that I made reference to was at Rutland House in September 1656 and was a private performance as musical performances were still against the law enacted by the Commonwealth which was not repealed until after the 1660 Restoration.
Your mention of the private performance for the King in 1683 was because he was a ruling monarch, not because it was then against the laws that were then on the Statute Book.
"Private" does indeed have a number of meanings.
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What is the viscous yellow fluid that both lubricates and nourishes cartilage in joints called?