I wish those that write this stuff would stop calling the guy a “safety pilot.”
There is no such rating.
The guy is the pilot, the rated pilot, and he was PIC, not the girl.
Yeah you are right...it gives the non-flying public the wrong impression...
In an aircraft, the instructor always has complete control (potentially). If the student is skilled, the instructor might not touch the controls at all, and a non-pilot might think of them as a "safety pilot". With a less skilled student, the instructor may do most of the flying. It is typical for an instructor to put hands on at critical moments.