I carry my regular wallet, with IDs, emergency contact info, etc. Sometimes I leave my debit card at home and carry enough cash for minor emergencies. I've considered getting a thinner wallet of synthetic material since my leather wallet gets damp when I carry it in my pocket rather than in the handlebar bag. But it's a hassle swapping stuff between wallets, and a risk of forgetting something important.

Regarding emergency medical care, there's always a risk anywhere that we won't be covered. When I worked for the federal government and had various HMO plans (Sanus, Cigna, etc.), I was surprised to discover my insurance did not cover anything other than major emergencies when I was on duty away from my home city. My job involved travel throughout north central Texas, and occasional travel to Chicago for training, but I wasn't covered for anything other than major medical issues, and even then only with prior approval -- which would be difficult to obtain in an emergency if I was unconscious or otherwise unable to call the insurance company for approval.