But you dont hourbuild w an instructor unless you want to...
There is no "safety pilot" coloum (!) in a FAA logbook, so to log it they need to log it as pic....... They are not logging PIC because they have command of the airplane, but because they are rated in the airplane....
Like if you have a multi private FAA, and do you ME-IR, you log PIC, because you are rated, but the instructor still have all the responsibilities.