To me, countable means capable of being put in a list. Thus, any set in bijection with $\omega$, or any finite set, is countable.
I think it's more about comprehension than definition. Any one is free to define anything they like (alá Smale), but are there good reasons for doing that?
The other thing to comprehend here, and which the OP indicates he does understand (but then why ask the question) is that an injection from set $A$ to set $B$ indicates that the cardinality of $A$ is less than or equal to that of $B$.
This is just personal preference, I guess, but I don't find the definition that excludes finiteness from countability as very interesting or exciting. That is, to reiterate, I really don't know who in their right mind would exclude finite sets from being considered countable. After all, countable is a descriptive and suggestive word. And it suggests that finite should be considered countable.
If you want to distinguish the two, there is always countably infinite.
But, this is all just my two cents. If you have some reason why you prefer it the other way, be my guest. To each his own.