Alternate names that have been or could be included in polls/surveys of citizens:
single-payer health care
single-payer national health insurance
Medicare for All
Expanded and Improved Medicare for All 

Clarification: The measurements of popular support among Americans very strongly indicate a desire for Improved Medicare for All when it is explained to them or it is asked of them in a clear manner. However, as of the summer of 2010, most Americans had never heard of Improved Medicare for All and/or had received enough propaganda against “single-payer” such that they were afraid of it at that time. We Americans must inform each other, especially during the years of 2010 and 2011. 

Notes are our personalized notes written on our letters flowing each month to the U.S. Congress, as part of the letters campaign: “Million Letters for Health Care”, for which you can sign up here.