Independent normal distributions

Question

I found two theorems with a similar content and want to find out which one is true:

Let $X,Y$ be normally distributed random variables and

1. $X+Y$ is also normally distributed or
2. $ (X,Y)$ is bivariate normally distributed,

then $X,Y$ are independent iff they are uncorrelated.

I don't know whether $(1)$ or $(2)$ is the correct requirement, but maybe they are also equivalent?