That is an example of a true gateway in use. You keep the customer on your site throughout the entire payment process. The customer enters the credit card info on your server and connects to the gateway behind the scenes for payment authorization. This requires an SSL certificate for security and you are responsible for record keeping and everything. The gateway simply authorizes the transaction and deposits the funds into your merchant account.
Many banks also insist on you having security protocols in place for your online business, such who has access to your databases, complexity of passwords, plus vetting and security monitoring of staff that have access to your online systems.
Using PayPal or Google tranfers the security side at least, directly onto the payment providers themselves. If there were multiple compromises of anyone's credit card data through a system malfunction, or dishonest staff, that could ultimately result in loss of credit card processing facilities. An extremely important consideration when considering payment options for a web site presence.