What is the difference between the two definitions that is not the following explanation: "Sell BTC 2.3 at $593.18" is an example of a limit order asking for $593.18 per 1 BTC. "Buy BTC 0.32 at $592.03" would be a limit order bidding for $592.03 per 1 BTC.
In a limit order, you specify how much of the asset (in this case BTC) you want to buy or sell, and the price you want. If there are matching orders on the book (e.g. someone who wants to sell at the same price, or lower, as the price at which you want to buy), your order will be filled immediately. If not, your order will stay on the book until matching orders arrive (which could be never).
(It is also possible for a limit order to be only partially filled, if the counterparty wants to trade a smaller amount than you did. In that case the rest of the order remains on the book.)
In a market order, you only specify how much of the asset you want to trade. Your order is then filled immediately at the best price currently available on the market. For instance, if you place a market buy order, you will be matched with the current lowest-priced sell order on the book. If that order is not large enough to completely fill yours, the next-lowest sell order will be used to fill some more of yours, and so on.