Two topologies generated by a partially ordered set - are they equal? Are they $T_0$?

Question

Let $P$ denote a partially ordered set, and consider two topological spaces with carrier $P$.

1. The topological space whose closed sets are generated by the upsets and downsets of $P$.
2. The topological space whose closed sets are generated by the principal upsets and principal downsets of $P$.

Are these spaces necessarily equal? And are either/both of them necessarily $T_0$?

Answer 1

A) No, they are not necessarily equal. Consider $(\Bbb R,\le)$. Now according to 1., the open intervals $(-\infty,a)$ and $(b,\infty)$ become closed, but not according to 2.

B) Yes, necessarily $T_0$, probably even more, already with the principal up/downsets: let $x\ne y$, then either $x< y$ or $y<x$ or they are incomparable. In the latter case or in case $x<y$, we have that $\{x\}\!\!\downarrow$ is disjoint from $y$ (hence its complement is an open neighbourhood of $y$ without $x$).