Reference: Why does the measure of a support $1$?
Billingsley - Convergence of probability measures p.70
In the proof, the author assumes the existence of a borel set $M$ such that $PM=1$. However, why does this set exist?
As you can see in my reference page, the measure of the support of $P$ need not be $1$.
I would assume that Billingsley is using a different definition of "support" than the one you gave in the other question. Specifically, a support of $P$ is just any measurable set $M$ such that $PM=1$. So, to say "$P$ has a separable support" just means that there exists such a set $M$ which is separable. (In the case that the entire background metric space is separable, then any measure has a unique smallest closed support, which is often called the support, and this agrees with your definition.)