I know with the Citibank online checking with the AA miles for the debit card that the bonus takes a while to post. I ended up calling twice before it actually posted. Since then, the miles post within 3 days of the closing statement on the account. No problems for the past 2 years.

As far as consumer protection goes, the same rights you receive with the credit card apply to the signature based transactions, as these are MasterCard based.

I prefer not to pay my bills with a pure credit card, instead, use the "credit card" feature of the debit card to rack up miles for the monthly, automatic payment of insurance, phone bills, cell bills, trash billing, water company and just about anything that does not have a transaction fee (like PG&E, mortgage companies, auto loan companies etc.)

Only one time did a I have a situation that the hotel did too high an "authorization" and the bank corrected it immediately. The few times that I have had an incorrect charged amount, the bank corrected it within the same parameters as a pure credit card. Usually, if you call the merchant, they will correct quicker. But that is MasterCard and Citibank, which seem to be more like AMEX in correcting problems, compared to VISA.