It really isn’t too broad — single payer, to reiterate, is where the costs of healthcare are paid for by the taxpayer. In Canada and the US single payer systems, this is true, but just with the middle man of the private companies. However, considering that their funding is directly from the taxpayer, everything they pay for (including infrastructure) is still government-funded. Further to that point, the NHS has been privatised quite a lot, with many parts of care being paid for by the taxpayer but provided by private companies.

Single payer is single payer, whether there is private sector involvement or not.