How is bartering (goods for goods) conceptually related to trading (goods for money) and monetary (money for money) exchange? My terminology may be off on this one but I can't get a definitive answer on how the above are related. To me, it seems like bartering is a superset of trading which is itself a superset of monetary exchange. 

The goal of every individual using money is to acquire things of value. Money is the means of acquiring valuables but of no value in themselves. Money indicates that at some point in time I produced, acquired, discovered etc something of value which became my possession but I later gave up receiving a numeric representation, which I deemed to accurately represent my possessions value at the time. I now have the "value" but not the valuable (effectively storing it for later use or subdivision).

Bartering is the exchange of something of value for something else of value (the ultimate goal).

Trading is the exchange of a valuable for a (perceived) representation of its value.

Monetary exchange is the exchange of two tokens of value accepted in different regions (or by different organizations, countries etc...)

It appears to me that the purpose of any exchange is ultimately between things of value (bartering), on top of which has been added an additional layer of abstraction (money) in order to have a constant (and consequently always desirable "something" for which to make an exchange) and abstracting the trade even further is to make an exchange between mediums of exchange.

Yet, most sources I view, trading is distinguished from bartering as bartering is an exchange between valuables but money is a medium of exchange. 
Is my assumption of a direct hierarchical link between the three forms of exchange correct?
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.