It is all in how you ask the question. OP got the correct answer to the wrong question.
If OP asked the question exactly as stated in his post, when he failed to show up for the rescheduled flight (or whatever reroute AA books him on), that segment and any onward or return segments would be cancelled.
Nonetheless, as AA cancelled OP's flight, OP was entitled to cancel the segment itself (and obtain a refund to the original form of payment). That would not cancel the onward segment. But, caution dictates having the agent confirm this after cancelling the segment and then double-checking the next day.
Bottom line is that one can't simply no show.