I agree with others that it probably would have been best to ask if the chairs belonged to someone else especially if they were in close proximity to the others sitting there. But, OP, I understand that sometimes we are not always thinking ahead or anticipating a problem. The lady certainly didn't handle this situation very well and it sounds like she embarrassed herself.
Also, she should have placed some items on the chairs if she was thinking that others in her family were coming down.
People can become so territorial. Off topic, a number of years ago I placed a blanket on a very crowded Ocean City, MD beach next to a guy and his companion. He blew a fuse. It escalated to the point that I called the police. I should have just moved but he was so ignorant I couldn't help myself in not wanting to move. It was afterall, a public beach. So, I agree that you handled things correctly by removing yourself from a situation that could get out of hand quickly.
Although, I understand how some feel that the chairs should belong solely to those guests paying to stay at the Poly, it is up t Disney to enforce or make this a rule. Until they do, I feel others are certainly not doing anything wrong by sitting there even if there are limited chairs.
Also, to the poster who just removes himself from people smoking in a nonsmoking area, that is your perogative. I don't think policing others every move is a duty of society (JMO) Equating that to ignoring a situation with someone being slapped, etc. is extreme. Everytime you confront someone in public you potentially put yourself in harms way. If I saw someone breaking the law or rules, I'd just report it if someone was available. If someone is being physically assulted, that's a completely different story. And, I would think most people would immediately call for help.