There are only two modes - an offline transaction or an online transaction. If the transaction is offline and you go over your overdraft or credit limit then your bank charge a fee and the interest rate sky rockets until you clear it. The bank still pay the merchant, it costs more for a merchant to process the transaction to avoid this kind of situation. Some cards like VISA electron are online only cards and can't be used on aeroplanes, some fuel pumps and by hotels without chip and pin (the ones that still stamp the embossed numbers!).
That isn't true at all that there are only two modes. Quoting Visa:
"Deferred Authorization occurs when an online authorization is performed after the card is no longer
available. The time delay may be brief, such as for a temporary communications failure or where the
merchant simply wishes to speed processing. The time delay may be extended, as when a ferry is out
of range of shore, for in-flight sales, or when the device does not have online capability (for example,
unattended kiosks where the transactions are offloaded nightly to a server and submitted in batches).
Merchants performing Deferred Authorization should complete authorizations within 24 hours of the
transaction."
Deferred authorisation is specifically DIFFERENT from offline authorisation. In offline authorisation, a transaction below a floor limit can be authorised by the card. This is what you're referring to where the bank still pays the merchant. In deferred authorisation, the card does NOT approve the transaction, and the merchant is liable if the card is later declined. Most Visa cards issued in the US are online-only, thus deferred authorisation is the recommended solution where online isn't possible.
See
http://technologypartner.visa.com/download.aspx?id=32 for more details.