Correct, although not all Publishing houses offer both options, but if they do, the songwriter has to make a choice.
The reason is simple. These organizations collect the monies due to the copyright holder, (usually, by contract, shared by the Publisher and the Songwriter), and belonging to more than one of the groups would constitute double dipping, (getting paid twice for the same song).
I do believe that the choice can be changed over time if the Publisher/Songwriter becomes disenchanted with their chosen PRO or SESAC chooses THEM as someone they would want to represent, (and that would be worth considering because they have the best reputation among the PROS).
Hope this helps,
JRH