$P_1$ is the set of all Problems such that every problem has an algorithm and nothing is defined about the algorithm.
So we don’t know anything about the algorithm. i.e. $P_1$ could be thought as set of all algorithms.An algo may or may not halt.
$P_2$ is the set of all problems that have an algo that halts. i.e. $P_2$ could be thought as set of all algorithms that halts.
So obviously $P_2 \subset P_1$