The FDA does not distinguish between artificial and natural preservatives. The FDA refers to preservatives only as ”chemical preservatives.” Unlike artificial and natural colors which have FDA definitions, the distinction between artificial and natural preservatives is made up by brands in order to facilitate claims that they have removed artificial preservatives. Here is the FDA definition of chemical preservatives:
The term "chemical preservative" means any chemical that, when added to food, tends to prevent or retard deterioration thereof...”
With regard to food packaging the FDA goes on to say:
A statement of artificial flavoring, artificial coloring, or chemical preservative shall be placed on the food or on its container or wrapper, or on any two or all three of these, as may be necessary to render such statement likely to be read by the ordinary person under customary conditions of use of such food.
Again, the FDA does not distinguish between types of preservatives. However, if you were to try and force a distinction between types of preservatives, you could probably claim that salts, oils and certain spices with preservative properties are “natural.” In fact, the FDA makes a point of excluding those things from its definition of chemical preservatives:
The term chemical preservative... does not include common salt, sugars, vinegars, spices, or oils extracted from spices...
Brands have coined the term “natural preservatives” in order to avoid the actual FDA definition of chemical preservatives, but they are not referring to salts, oils and spices and instead have somewhat arbitrarily designated certain chemical preservatives as “natural” when clearly they are not. Since they can’t say that they have no chemical preservatives, they claim that they have no “artificial preservatives” instead — which is false.