BA will zero out D class on some routes at the drop of a hat whereas AA is more generous with mid-tier business availability. On the other hand AA has recently started zeroing A class on many popular domestic routes to discourage upgrades, notably to/from Hawaii, which is unfortunate for anyone buying international A tickets on a regular basis (AONEx or otherwise). Meanwhile LA and JJ often have only full Y availability on domestic routes but (according to reports on FT) if revenue management find out you're on an LONEx fare then L space will be found for you.

Note that availability may also differ on point of sale. For instance if you phone the BA office in New York you will usually get less access to the cheap fare buckets (I/R/D) than if you phoned Amsterdam, for the same exact flight.

As for BA getting the rules wrong ... unfortunately many BA telephone agents are clueless even at the best of times. Don't believe everything you hear. But some of them are really excellent. Good luck.