Connected but not adjacent vertex
You've exactly described the first situation, i.e. we indeed say that $u$ and $v$ are non-adjacent. For the second we do the same, but we want to also specify they are in the same component of $G$ which contains them if $G$ is not connected. 

These are already simple and well-understood descriptions and there's nothing "more standard". However, you are also free to use your own definitions in your actual use case if it's worth it.