Definition:Countable Set/Also defined as

Definition
Some sources define a countable set to be what is defined on as a countably infinite set.

That is, they use countable to describe a set which has exactly the same cardinality as $\N$.

Thus under this criterion $X$ is said to be countable there exists a bijection from $X$ to $\N$, that is,  $X$ is equivalent to $\N$.

However, as the very concept of the term countable implies that a set can be counted, which, plainly, a finite set can be, it is suggested that this interpretation may be counter-intuitive.

Hence, on, the term countable set will be taken in the sense as to include the concept of finite set, and countably infinite will mean a countable set which is specifically not finite.