You are correct that the classifications are not established by American Express. My (primitive) understanding is they are established by the credit card processor that the merchant uses, and there doesn't seem to be any meaningful incentive for the merchant to ask the processor to correct them.

To see the classification of a particular transaction:

Go to http://americanexpress.com
LOGIN
Select the appropriate card (if you have more than one)
Click Recent Activity -or- Online Statement
Click + to display details of a specific transaction
Click More Detail
In Transaction Details window look at Merchant Type:

You can print the window if you wish, but I think it will be awkward to approach a restaurant to complain because you could receive a larger rebate if the establishment were classified differently. Logically, the owner should want to maximize your rebate if it does not increase the transaction fee, but in some instances I suspect [SPECULATION] the processor has manipulated the classification to reduce the fee [/SPECULATION].