A "processing system" refers to a set of activities or operations that are carried out in order to handle or deal with something. It is a way of organizing and handling tasks or information in a systematic manner. Full definition
The first one is for clients who have no payment processing system in place. (links2exchange.net)
I have taken the responsibility of operating and monitoring small as well as large scale electronic data processing systems which helps me to take decision on critical tasks on both type of systems. (exforsys.com)
Now people can pay with their smartphones or watches and it's imperative that you have some type of credit card processing system in your business. (think-creditcards.com)