I have not been successful in finding an online company that handles
credit card processing completely online FROM THE BUSINESS-OWNER'S
POINT OF VIEW.

See, there are tons of companies out there that handle online credit
card processing for let's say Amazon.com or whatever.

But what I want is a company which allows me to receive a few
information about my customer (only their card type, card number, date
of expiry, and perhaps card-holder's name) and then log-on to MY
ONLINE ACCOUNT and by punching in those info, charge them.

In a nutshell, what I want is an online service that does what
phone-driven credit-card orders require. (without the need for any
hardware installed)

Please list a few of these services and tell me what your own
recommendation is. It would be better if in addition to the
functionality mentioned above, the service also handles the
amazon-type kind of credit-card processing (done by the customer).

Thanks and please let me know if you need some clarification.