Ian Golding takes the time to discuss the differences between the definitions of several approaches regarding customers. He uses customer experience, customer service, and customer-centric. It’s a good article on the subject and it culminates in his definition of a customer-centric organization:

Putting customer focus at the heart of everything you do, in order to achieve customer satisfaction and loyalty

I’d agree, and it’s essentially how I would define customer success. When customer success is a vital function of your business, your Success team is there is ensure that customer focus is at the heart of everything you do. Or, as I paraphrased Kathy Sierra:

Customer success is about a subtle shift from focusing on making excellent products to making excellent users.

When your focus is there, success for your own company will follow naturally.