I think you are overthinking this question.
The price of a crypto-coin is whatever price it gets traded for. The value, or "market cap", is the current price of one coin times the number of coins in circulation. (or milli-coins, or ethers, or Satoshis, or whatever the currency's unit of measure is called)
A cryptocurrency exchange is a place where people can post ads. They can say, "I want to sell 5 bitcoins for \$1000 each (or more)." or "I want to buy 4 bitcoins for \$900 each (or less)."
When two ads match up - when someone says "I want to sell 5 bitcoins for \$1000 each (or more)." and someone else says "I want to buy 3 bitcoins for \$1100 each (or less)", then they get matched with each other, and 3 bitcoins are moved, and the price is somewhere between \$3000 and \$3300 (I think it's typical to use the price of whichever person posted their ad first). Then the first person's ad is automatically updated because now he only wants 2 more bitcoins instead of 5.
You can also post an ad saying "I want to buy 5 bitcoins for whatever is the best price right now." and you'll match up to whoever is selling them for the lowest price. Or "I want to sell 5 bitcoins for whatever is the best price right now." and you'll match up to whoever is buying them for the highest price.
And then the price of Bitcoin is just the price of the last trade that happened.