Single payer in the US simply means the patient doesn't front any money. The doctor/hospital/... just gets the money for medical services rendered directly from insurance (and insurance collects things like a deductible from the patient, not the doctor. Essentially, insurance guarantees payment to the medical provider and deals with all patient financial matters). Medicare and medicaid are single payer.

Things aren't called the same thing everywhere. For example, "single payer" is called "third payer" in Northwest Europe.

Expensive (e.g. expat) insurance tends to be single payer/third payer even though it is 100% private insurance. You just go to a doctor, they help, other than your insurance card, no questions asked.