In the US, you're probably referring to the ACH system (as mentioned in the article you linked to). This is the same system used to clear checks, so in essence - direct credit is a paperless check. It takes at least 1 business day cycle for this transaction to clear (sometimes longer, depending on the chaining between the involved entities, as the transactions are processed in nightly batches).
Wire transfer, on the other hand, is a transaction transmitted separately on-line (or in much more frequent batches) through the Federal Reserve wiring system (or, internationally through SWIFT system, or IBAN system where supported). This transaction typically settles within hours, is not reversible, and once settled - is considered as a cash deposit (as opposed to checks, that even after being settled may bounce, get cancelled, or be discovered as fake - and the money will be withdrawn back out of your account).
When you tell someone to wire you money - you would probably expect a wire transfer, not a ACH deposit.