Pyrate is a variation of the spelling of pirate. Spelling during the 1700 and 1800's was haphazard and there were many ways to spell pirate, with pyrate as one of the most commonly accepted forms.
Pyracy in the 1700-1800's was defined as the act of theft on the high seas (out of the jurisdiction of any country). Pyrates could be captured and tried in any country and were subject to any sentance laid down by that country.
Henry Morgan was considered one of the most successful pyrates of all time. He attacked many French and Spanish ships illegally and when he was captured by the English, the Queen knighted him and made him admiral of all the English fleets sailing the Caribbean.