I wish these stories would stop getting aired, but I have to say something, looking at the photos....
I want the merchant copy with their signature, not the customer copy with what they wrote. The customer keeps the customer copy. They could have wrote anything, and the ink looks different on both copies. Somebody's fibbing, but I am more likely not to believe it's the family just because I don't want to believe anyone is really that rude.
Edit to add: Saw the video. If their credit card statement is showing the amount written and she's telling the truth, it may be a case of there's a third party, a colleague who tried to harass her. I find that far more likely than customers who receive good service leaving nasty notes to their servers, or anyone doing that, really.