First, we need to define terms properly, which is much harder than it seems at first, and too seldom done. As Thomas Sowell says, for example:
The biggest of the big lies in the "health care" hype is that a lack of insurance means a lack of medical care. The second biggest lie is that health care and medical care are the same thing.
"Health care" is simple enough to define - any action, taken by an individual or another party, that affects the individual's health. If the definition sounds broad, it is; try thinking about something you do which never has any effect on your health.
"Medical care" is a bit tougher to define exactly. I think it works to define it as professional services rendered by doctors, nurses, any medical specialists(including but not limited to opthalmologists, oncologists, urologists, orthopedists, Lasik surgeons), paramedics, EMTs, first responders, and any similar person, performed generally with the expectation they will be paid for their services. (Some EMTs and first responders are volunteers, but this is uncommon and even in this case the EMS service/fire department will often bill for medical supplies and transport expenses.)
"Health insurance" is tougher yet because the term usually describes far more than is meant by "insurance" in any other area, namely the payment of a premium in exchange for the guarantee that a third party will pay a specified sum in the event of certain nonroutine losses/expenses. For reasons to be mentioned below, the U.S. tax code has given a strong incentive for routine expenses such as prescription drugs, annual physical checkups, and dental cleanings to be included under the rubric of "insurance", whereas for example car insurance does not commonly cover an oil change or a trip to the detail shop.
I think for my purposes of explanation, health insurance can be defined as any contractual system by which medical care for a given patient is directly paid for, in full or in part, by a third party(i.e. a party other than the medical professional and the patient). This can include systems like Medicare/Medicaid which are paid via tax revenues, employer-based coverage, or personally purchased policies; but the most important and relevant detail is the third-party payment system. The incentives for conduct of patient and doctor are much different when a third party is paying, as opposed to when the patient is paying out of pocket, and this is the fundamental source of administrative complication in modern medical care.
Next to come - a brief history of modern medical care.