Sorry I probably missed "$(P_t)_{t \ge 0}$ of a Hunt process" when I first composed this answer and that is why confusion arise.
I think it is most beneficial if we start from and stick to definition.
Hunt process is a strong Markov process with some regularity on sample functions, usually càdlàg sample functions but subject to some alternations. Suppose $X$ is on a (locally) compact space $E$ and there is a one-to-one correspondence between Hunt process and Feller-Dynkin semigroups.
The paper you cited provided (1)(2) as a set of equivalent conditions that allows you to regard the transition semigroup $(P_t)_{t \ge 0}$ as a Feller-Dynkin semigroup as well. A Hunt process may admits a transition semigroup satisfying (2) of course but it also needs (1) to make the operations within the transition semingroup closed.
I hope I made myself clear now.