All of the examples I gave had a proper UI. Automated customer support doesn't have to be delivered via a chat or voice interface.
Just to clarify, when you said you used automated customer support to "cancel orders, cancel subscriptions, change a broadband package, arrange a return/refund for faulty goods", you're saying you used UI to do those things?
Correct.
The goal of communication is to share ideas and reach an understanding, not to be technically correct. No one else was thinking about websites when talking about "support" and I'm guessing you knew that.