One of the official definitions of countable set is a set $A$ such that there is a bijection between $A$ and a subset of the natural numbers. That definition includes the finite sets.
If a set is countable but not finite, it should be called countably infinite.
But people often sloppily write "countable" when in principle they should write "countably infinite."
So if the phrase is in a problem on an assignment, you may have to ask the instructor. However, the answer may be clear from the context. And if, for example, you have shown that a set is in one to one correspondence with the natural numbers, you will have shown that it is countable. For you will have proved the stronger result that the set is countably infinite.
As to the second question, "countable" and having "countably many points" mean the same thing.