Definition:Countably Infinite/Also known as

Definition
When the terms denumerable and enumerable are encountered, they generally mean the same as countably infinite.

Some sources use the term countable to mean specifically countably infinite, that is, not including finite sets as countable.

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

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

However, this definition also seems to be going out of fashion, as the very concept of the term countable implies that a set can be counted, which, plainly, a finite set can be.