A socialized health care program that provides a standard of care to every resident is single payer. Full stop. And almost every industrialized nation but us has exactly that.
If that standard of care can be augmented with superior care through a private program as well, that doesn’t discount the fact that everyone still gets a standard of care that is paid for and controlled by a social system. That’s still single payer. If you can get care to a broad array of medical issues, supported by a public trust, and it costs you nothing at point of service, that’s single payer, regardless of how you might want to split that hair.
So, if the government system will pay for most office visits, most hospital stays, most home health care and most mental illness treatment, but perhaps won’t pay for experimental treatments and surgeries, and exempts some classes of pharmaceuticals, for instance, that’s still single payer. Everyone gets that level of service based upon our tax levies.
What you are referring to as single payer is a completely government controlled medical system in which access outside the system isn’t just unavailable, but illegal. Based on that exclusive, restrictive definition, you’d be right. But no reasonable person uses a definition that is so restrictive.