I never understood this part, I have to be honest, whether "distribution" is countable or uncountable. A
distribution, being a
function, or an
object if you wish, is parameterized by parameters. They can be real numbers (uncountable), or integers (countable).
My understanding is that an English speaking person would see the Poission distribution, which is parameterized by one real number, as an uncountable object (like sugar, water, etc). Real numbers cannot be counted, so I would expect that the set/class of all Poission distributions is uncountable. It should be either with the or without it.