The formulation I prefer: a countable union of counted sets is countable.

(Any countable set has a well-ordering, because a bijection with the natural number gives you one in an obvious way. The trouble is that you need well-orderings for all of them together. The unusual term "counted" emphasizes that we need the actual "countings" to do the job, whereas for me "well-ordered" is sufficiently commonplace that it doesn't shove in my face the requirement that each set come along with a specific choice of well-ordering.)

In my mind, "well-ordered" is fully distinct from "well-orderable" :) but "counted" is unambiguous, you're right.

As Lang used to say the Axiom Of Choice is obvious, 'I just pick the elements'. Just kidding, thanks for this interesting logical point.