Sorry but... I have to disagree too. The customer is not always right. I understand where you are coming from with your stance that it is your money. but really it does only take 2 seconds to add on a hello. In any business, one would hope to be treated with basic common courtesy. If I am at work and receive a call, once I pickup and say hello, I certainly wouldn't expect a client to instantly just say "I need you to tell me about XYZ". (In fact, it would put me off as well!) I would hope they would at least say "Hi Hopeful, this is [Name]." before jumping into a question or request. It sounds silly and stupid, but a little common courtesy goes a long way.
I understand that you're paying for the time, and that every second counts, but whether on chat or phone, a quick hello is by no means going to make that big of a difference, especially if you already have a copy/paste question scripted and ready to go. Her response may seem extreme, but all she is asking for is to be treated like a person. She will say hello and ask if you've gotten the free minutes she has sent because she will send free minutes to make sure that she connects first so that she does not waste your time (which is why a lot of the time she will end the chat first, send you the mins, and once you get back in she asks if you have received them. Or from what I've heard from others if you email her first she sends them and then makes sure that you got them.) She's being decent and polite making sure she isn't wasting your time with those free minutes, and in return is just asking for common decency, that really isn't costing you a dime.