Plagiarized from the wikipedia article, I can tell
Actually I think it was some dictionary site, but definitely from somewhere. No use reinventing the wheel.
So here is where you're wrong. Well, not wrong about how markets work, but you are wrong in that market exchange is only one kind of exchange. There are kinds of exchange that have absolutely nothing to do with the market. You are concerned with only one particular type of exchange (buying and selling, or exchange for gain) that is what we use "markets" to describe, but exchange is much more broad than that. Your parents giving you presents for your birthday is a form of non-market exchange. The government giving you a social security check is a form of non-market exchange.
This is why I said you should take a look at some of the ethnography of societies that are very different from our own. You will see all kinds of exchange the goal of which is not "to get a good deal".
Some good examples would be the ancient civilizations of Egypt and Sumer - these were highly complex literate societies where buying and selling goods played almost no role in the economic system. Instead hugely complex bureaucracies developed around palaces and temples, and these administered rationing systems where equivalencies between goods (a weight of silver and an amount of wheat, for example) were determined by edict rather than by buyers and sellers interacting in a market.
You seem to be assigning me something that you can knock down for some reason. Your limitation that only "buying and selling" is "exchange for gain" is unique to you. ALL exchange is "for gain," or it wouldn't happen.
The ancient societies you describe had an economic system that regulated a market...and one of the "goods" that was exchanged in that market was the loyalty of the people to the governing body. The good that was most commonly exchanged for that good was food. The food was generally acquired from surrounding landholders in exchange for protection. This market worked quite well in its time, and I'm already familiar with it, so I see no need to "study."
On your other examples of "non market exchanges"...
Gift giving is NEVER done without expectation of return. It may not be negotiated return, and may sometimes result in disappointment, but it is certainly just as much a market exchange as any other.
Social security checks are nothing more than the talent of bread handed out by the ruler of Babylon.
A lot of what you might think of as "non-market" are actually just misleading because people think of the
spot market. Continuing delivery of bread, or eventual delivery of SS checks, in exchange for lifelong loyalty as a citizen is not the "this in hand for that in hand" exchange that is easily followed in a spot market, where delivery on both sides is immediate upon completion of negotiations, but it is still an exchange, and it is still governed by the rules of the economic system, and it still occurs in "the market."