Check you Visa agreement. You're not liable for charges made on a stolen card.

FWIW I had a New Zealand credit card stolen in the US in February. Remarkably the crook managed to rack up almost $1000 in small transactions over two days at gas stations all over town. While the CC issuer credited the $, it was a real inconvenience. Now I have a new CC for the US with the signature replaced by a "please request ID" label and a stripe that converts any attempted use at an automated fuel bowser or whatever into a "see teller" request.

FWIW the local (US) police tell me that refusing to show ID may result in greater loss of privacy because of retention/extra attention paid to mugshots taken by the hidden cameras all over supermarkets and retail outlets these days. Face reading technology, merging security companies and billions of labelled online mugshots (including on UT) means all these things will be linked up over time and you'll be far more "visible" than the rest of us. ;-)