Am not looking for precise definitions of the differences between private healthcare vs PHS i.e. public healthcare systems.

• funding — PHS is funded by tax payers. I think private health system is funded by revenue and possibly donation. University healthcare infrastructure might run on university’s endowment.
• administration — PHS is run by government departments. Private means private institutions including government-owned firms.
• fees — patients pay zero or low fees (polyclinic, MYS) to access PHS
• waiting list — given the low fee, service users far outnumber service providers. PHS usually have some kind of prioritization criteria. In private, fees are the natural criteria.
• doctor compensation — Simple in private case. In PHS, doctors need to be paid adequately but presumably lower, with a higher workload. I would assume some doctors take it as public service and civic duty.
• advanced diagnostics + treatments — private doctors would use (or recommend) the latest and most expensive options. PHS may not have the same options easily available.

— insurance is a complicated matter

By and large, I would say insurers are commercial (private) companies, whereas many aspects of the operation are regulated by governments — premiums, exclusions, billing, compulsory enrollment,

Many countries’ health insurance is largely run by the government, negotiating with domestic/foreign commercial insurers.

In some cases, a medical insurer is government-owned or government-linked, but I believe every insurer must endeavor to make a profit to be sustainable. Insurer is never a charity. If run carefully, the health insurance business model is sustainable. The members who seldom claim from the “pool” would effectively subsidize those frequent claimers.