In FAA-land that is correct. You may not actually be the PIC, but can log time as PIC (FAR 61.51 (e) (1) (i) .. When the pilot is the sole manipulator of the controls of an aircraft for which the pilot is rated), and the FAA has confirmed that rated means rated to fly the aircraft at all, so basically after you get the PPL, everything is logged as PIC if you actually fly the thing.
Peter wrote:
However in UK-CAA-land, every flight with an FI would have the FI as PIC
That, however, is not true. If somebody asks me to fly as safety pilot, there is a simple question – “do you want me to teach, or just be along for the ride in case something goes wrong?”; and if the answer is the latter, I point out that they are PIC, I will only do something if they ask me to or if I think we are about to die. I know that @Timothy prefers death and does not do the “intervene if I think I am about to die” bit. 
That does not prevent me from giving feedback after the flight, though.
I am very, very clear to the PIC that if I am a passenger, I am a passenger.
OK, yes, I didn’t mean to say that everybody who has an FI rating is the PIC, because that would preclude such a person flying in the RHS as a passenger, and you would get into this stuff.
Rather, I meant that in Euro-land, if an FI is acting in an instructional capacity, the other person is always PUT. I have seen both PIC and PUT logged in this scenario. The FI has not great reason to care how you log it.
Timothy wrote:
I am very, very clear to the PIC that if I am a passenger, I am a passenger.
But if I’m a safety pilot, I’m a safety pilot… and I want to be clear what that means, i.e., I am not PIC, but ready to take over when asked or when we are about to crash.
I completely understand your approach – it just isn’t the same as mine…
Anyone reading this should be aware that Timothy has VASTLY more experience doing this than I do…
To support @boscomantico, also PUT is no EASA term, in the described scenarios one can either be PIC or log the time as DUAL, according to current regs. Unfortunately EASA leaves it to the competent/incompetent authority to precise the exact handling, e.g. German CAA issued NfL II 330/17 which clearly states that even club internal checkouts, which are not needed according to licensing will have the instructor as PIC. And in the NfL there is also a strict only one PIC rule mentioned. So every CAA might have a little different opinion about this.
