It should not still be needed to say:

The notion that some infinite sets are countable is a special mathematical notion. It is technical and has a rigorous formal definition. There is no layman's version of it.

I may not be a mathematician but I can take care of my own bank-account via numerations of various sorts just fine — javra

No one doubts that you are a grown up person who can do numerical reckoning just fine. The point is that you don't know anything about the mathematical notions that you have referred to in this thread. And I don't mean your philosophical notions, but rather the specific mathematics you have also referenced. Being able to balance a checkbook is nice, but it's not an informed understanding of the mathematics you've commented on.

df. x is countable iff (x is one-to-one with a natural number of x is one-to-one with the set of natural numbers).

As far as I can tell, that is different from the everyday sense, since the everyday sense would be that one can, at least in principle, finish counting all the items, but in the mathematical sense there is no requirement that such a finished count is made.
— TonesInDeepFreeze

No, not in my neck of the woods. The everyday sense would be that one could, in principle only, count an infinite series of elements/units/items for all of eternity yet to come and still never get to finish. — javra

Your neck of the woods is a fantasy place. People in everyday life don't take 'countable' to mean "one could, in principle only, count an infinite series of elements/units/items for all of eternity yet to come and still never get to finish."

I'd like to see just one person at your local supermarket in your neck of the woods who says anything like, "My understanding of what 'countable' means? Oh, it means that in principle you could count an infinite number of things for eternity but still not finish."!