Yes, that's right.
Perhaps someone else can confirm, but I think the advantage of reprinting on AA stock (though it's not necessary) may be that the AA BP will say "DOCS OK" and perhaps you can then skip the docs "interrogation" just before the gate, which can be a hassle/bottleneck.
One of our party had a mini-"interrogation" at the Admirals Club in T3, whereupon they put some sort of sticker on her BP wallet. They didn't explain the sticker but indicated that it would be helpful somehow. The sticker later seemed to satisfy the LHR security doing the docs "interrogation," who kind of waved her through the "interrogation" upon seeing the sticker. Maybe that sticker is equivalent to a "DOCS OK" BP, but that part was not explained to us.