Another ideea for a coin is to calculate the trade price (estimated) based on transactions, and this to be done from the main server. This is just to keep it at a real/fair price for the entire market.
But this doesn't mean that exchange websites can't apply their own price.
You seem to have a very strange idea of how prices work. Exchanges do not set their "own" prices, the users set prices for themselves. The exchange site has a matching engine which searches all the orders placed by the users, and if finds someone willing to sell for a price that someone else is willing to buy at, it makes the trade. A trade will not happen unless the price is agreeable to both the buyer and the seller, and this is true for all
transactions, not just those that take place on exchanges. No other price is "real" or "fair".