Unfortunately the rules change every few years, which is why you are seeing conflicting information. The AA.com website isn't too helpful either at the moment. Last I knew the "official" policy was battery powered devices only, but the website says nothing that I can see at the moment. Best to contact AA Special Services and ask.
That being said, I've never had any trouble using my CPAP on board an AA flight without a battery. I always call Special Services before departure to advise them I'll be using it, and they always warn me not to depend on the availability of at-seat power and not to use the CPAP during take-off or landing.