Your ticket number is a 13 digit number that starts with 001, which means AA issued it. You will very rarely actually need this number, though you do want to make sure it exists. There is no partner ticket number. If CX wants your ticket number for some reason, use the one you have starting with 001.

Your PNR/record locator is a 6 character alphanumeric code. You have one for AA and a different one for CX. You want the CX one to manage your flight on their website. Call AA or use Tripcase to get the CX one.