In Smith's theory, what is the difference between the "nominal price" vs. "market price" and "real price" vs. "natural price"? In Adam's Fallacy, Foley writes: 

"Smith distinguishes what he calls the nominal price of a commodity (the amount of money for which it exchanges) from the real price of the commodity, the amount of labor required to produce it." (p. 12)

Later he writes: 

"The market price is just the amount of money for which the commodity changes hands at any particular moment; it rises and falls because of shortages and gluts, changes of taste and supply, and speculation. But Smith believes there are important forces that tend to push the market price back toward a certain level, which he calls the natural price of the commodity. Since there are always disruptions in any market, he does not expect the market price to converge smoothly to the natural price and then stay there, but instead believes it will fluctuate or (adopting an image from Newtonian planetary physics) “gravitate” around the natural price." (p. 13 et. seq.)

Is there a difference between the nominal price and the market price, and between the real price and the natural price? Because to me, the two pairs of terms seem to match up.
Adam Smith in his writing defines the nominal as the price of something at a given time counting prices out of market equilibrium. He refers to it as the price at what something is exchanged between two individuals not taking into account general market conditions or particular affairs at given time. 

It's just the price in monetary terms at what something it's exchanged and can or cannot represent the fair trade for one or both parties at a given period.

Market price for Adam Smith is the price at what something is generally traded in the market at given time. It is different than the nominal price and sometimes it's rare when they are the same but nominal price should tend to market price if certains conditions are met in a most free and untroubled markets.
Adam Smith also writes about free market irregularities, monopoly in labor , capital or land markets. 

He also makes an extended analysis and compares a market price for gold and silver and how it is different in terms of time periods or given countries due to; coin making fees, coin metal compound, weighting devices and standards, easyness of purity verification or just shortage and abundance of metal resource among others.

The real price he writes is what something costs to produce purely by it's labor value required to obtain it, via exchange or by producing it. I quote "The real price of everything is what anything costs for the person whom wants to posses it, the effort and the fatigue that his adquisition requires." 

The natural price for Adam Smith is the price that justly covers the average cost rates of, land leasing, labor wages and capital gains destined to produced it and sell it on the market, and nothing more. Then he writes we could say something is exchanged by it's natural price when that price represents the minimum or cero gains for the factors required to produce it.

I make the observation that those are the 3 main factors that make posible the exchange of a product or service in terms of the classical economic view established by Adam Smith and his contemporary collegues.