What's the difference between 'fowl' and 'poultry'?

As the title says, I'm interested in the semantic difference between these two words.

OALD gives the definition of fowl as "a bird that is kept for its meat and eggs, for example a chicken."

Poultry is defined by the same dictionary as "chickens, ducks and geese, kept for its meat and eggs."

While the two definitions sound quite identical to me, my understanding of the difference between them is that fowl is more narrowly focused on a bird being treated as a single object, but at the same time allows for the flexibility of the semantic content of the word, so that the word is not rigidly associated with and applied to any specific kind of birds.

Poultry in turn seems to treat the object in its plurality and also to have some minor constrictions on its semantics, thus limiting a wide notion to a number of particular cases.

With all that said, I ask that you either confirm or deny the validity of my intuitive explanation for the question I'm seeking the answer to.

I think there's another difference: poultry is a subset of fowl. Either poultry or fowl can refer to domesticated or farmed animals. When referring to wild game, however, (e.g., pheasant, partridge, wild turkey), fowl is more applicable, whereas poultry is not.

Merriam-Webster's defines as follows:

poultry: domesticated birds kept for eggs or meat

fowl: (1) a bird of any kind (2) any of several domesticated or wild gallinaceous birds

Interestingly enough, my online dictionary says that gallinaceous includes "domestic poultry and game birds."