In https://en.wikipedia.org/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent, it states
To say that the pair $(X,Y)$ of random variables has a bivariate normal distribution means that every linear combination $aX+bY$ of $X$ and $Y$ for constant (i.e. not random) coefficients $a$ and $b$ has a univariate normal distribution. In that case, if $X$ and $Y$ are uncorrelated then they are independent.[1] However, it is possible for two random variables $X$ and $Y$ to be so distributed jointly that each one alone is marginally normally distributed, and they are uncorrelated, but they are not independent; examples are given below.
What does the "to be SO distributed jointly" part mean?
This is a grammatical error on Wikipedia. It should read (in my opinion):
The statement is trying to say that if $X$ and $Y$ are defined on the same probability space, so that they have a joint distribution, $\mathrm{corr}(X,Y) = 0$ does not imply $X,Y$ are independent.
This is important because of a subtle technicality with normally distributed random variables. If $(X,Y)$ is normally distributed (that is "$X$ and $Y$ are jointly normally distributed") then $\mathrm{corr}(X,Y) = 0$ implies $X,Y$ are independent. However, if $X$ and $Y$ are both normally distributed, it is actually possible that $(X,Y)$ is not normally distributed, in which case $\mathrm{corr}(X,Y) = 0$ does not imply that $X,Y$ are independent.