The Notification Center seems to be missing this morning - I hope it come back soon!
But can someone explain to me how texts work in the US? I though you just sent text messages to the cell phone number, so what is all this about specifying the carrier?
In any case, I am guessing that AA do not want to incur the expense of sending texts to phones outside the US. So I put my phone's email address in the field, which seems to have worked, as when I create a flight status notification from the reservation, it shows the email address in the field there described as "Deliver to: Device email or text message address:"
(Note most cell phone operators provide an email address for cell phones of the form <number>@<operator>, which converts the beginning of the email into a very short text message. This is a good workaround to get text messages from aa.com on non-US cell phones.)