Elgin's question brings up an interesting angle. The quick and easy answer, Elgin, is that Nova Scotia in those days included what is now New Brunswick. The division into two provinces happened around 1785, partly in response to the 1783 influx of Loyalists especially onto the coastline of the Bay of Fundy that is now in New Brunswick. The new home of these passengers of the Duke of York was called Nova Scotia when they got there in 1772-ish, but within a dozen years or so, it was New Brunswick.
But the arrival of the Duke of York in Halifax is interesting. Thanks, Elgin, for bringing it up. Did the ship then carry on around the south-western tip of Nova Scotia, current day Yarmouth, and bring its passengers into the Bay of Fundy to drop them off near their land grants? If not, how did they get from Halifax to their eventual settlement? Stu, I'm assuming you will know this, so i'm not even googling it.