How is bartering (goods for goods) conceptually related to trading (goods for money) and monetary (money for money) exchange?
Consider an economy where there are $N$ goods and assets. How do you purchase a good? If $n \cdot (n-1)$ forms of exchange are possible because all $n$ goods can be all other $n-1$ goods then this is a barter economy. If there is only a single good that can be exchanged for all other goods then only $n-1$ forms of exchange are possible then this is a monetary economy where that single good is "money". See, for example,  Kiyotaki and Wright (1993). Used this way, Trade Equilibrium $\supset$ Barter Equilibrium $\cup$ Monetary Equilibrium. A no trade equilibrium is sometimes called autarky.