BA.com is right on this occasion. You'll be booked into a 'business' booking class on the BA flight numbers so will only earn business TPs. The AA site says First because the cabin is actually (domestic) First, but BA will award the points based on what you booked.

If the extra TPs will be valuable to you, I'd advise you to call BA to cancel the booking under the 24hr rule, then rebook under the AA flight numbers (in P/A class) either on AA.com or over the phone with BA. The price *should* be the same but check the availability before you cancel.