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Neither -- a hit batsman is a pitch thrown that hits the batter, even if the catcher immediately catches the ball after that contact.
A wild pitch is a pitch thrown so badly that the catcher is not able to catch it, but which does not hit the batter.
An "error" in baseball terminology has nothing to with pitching -- it is a fielding mistake. A pitcher can commit an error, but only on a batted or thrown ball.