The way that this would work, if AA were serious about it, is this:
1. You put reservation on hold at AA.com.
2. You call in with details of lower fare.
3. They check out lower fare while you are on the phone.
4. They adjust the price of your reservation to be $50 less than the fare at the other site (you are still on the phone).
5. You purchase ticket online.
This eliminates all the hassle of "we can't find it" and "that fare doesn't qualify because we put the exception in the tiny print".
Obviously, AA doesn't want to really offer its lowest fares on AA.com, it just wants you to think they do.
If I'm wrong, since we know that AA reads Flyertalk, they will change their guarantee over to something very like the above, and I will be back here with my apology. Until then, I will continue to check at least 5 other sites for every fare, and book it where I find the lowest price.