See my musing on objective and subjective value for more on this subject

I respectfully disagree. It is precisely because both traders place different values on the items being exchange, that trade *ever* occurs. It's nonsense to suggest that I would trade X for something that I valued exactly the same as X (or that someone else would be the counterparty). Mutually beneficial trade occurs when I trade my X (which I value less than your Y) for your Y (which you value less than my X). Sometimes, of course, our estimations turn out to be mistaken, but this is by no means a requisite part of "exchange".

1. He can sell his invention to insurers a make a handsome trillion dollars, a small price to pay I suppose for the eternal alleviation of human suffering.
2. He can give his invention to the world and live off of his goodwill and reputation (probably not a bad life, either)

Most likely, he'll be assassinated by a cabal of other doctors.

A free market is a market comprised of voluntary exchanges. A voluntary exchange only requires two people, who may use money or not, as they choose.

I have already shown that prices do not reflect any objective value, because the price is whatever the buyer will pay, which is purely subjective.

What is the price of a sack of corn to a farmer? I take an example from:

The farmer needs the first sack just to survive until the next harvest. The second sack will ensure he eats well. The third sack he would use to feed poultry to provide variety in his diet. The fourth sack he would use to make whiskey. The fifth sack he would use to feed pet birds.

In these circumstances, the value of the fifth sack of corn is low to the farmer (although not to the birds). If he lost it, he wouldn't pay much to replace it, if at all. He would just stop the use that provides him with the least value–feeding his pet birds.

As the farmer loses each successive sack of corn, the value rises.

Where is the contradiction between our two definitions of capitalism? And how does your definition exclude the exchange of a back rub for a hair weave?

Well, first of all, Capitalism needs money to work.
Without it, you are only exchanging items of equal value to fulfil our own desires. Thus, nobody profits unless by miscalculation.
In your example, we are tradin a Commodity for a Commodity. Simply C – C.
When you put money in the equation, your example becomes C – M – C. Thus you trade you commodity for an equal value of money and use those money to buy my commodity.
A Capitalist works in this equation. M – C – Mi, or Money – Commodity – More Money.

His objective is not to get an item of equal value but to get more money.