Denoting phase gate by $P$ probably comes from its name Phase

Phase gate is ussually denoted by $S$ as you can see from Nielsen and Chuang book example. Moreoer, symbol $S$ is used also on IBM Q.

To sum up, it is a matter of a paper's author decission. As I remember, I have seen phase gate denoted by $S$ only.

Denoting phase gate by $P$ probably comes from its name Phase

Phase gate is usually denoted by $S$ as you can see from your Nielsen and Chuang book example. Moreoer, symbol $S$ is used also on IBM Q.

To sum up, it is a matter of a paper's author decission. As I remember, I have seen phase gate denoted by $S$ only.