I actually see that I slightly misread the question that Ron answered "Correct" to. I agree with his answer to the question actually posed: can the sole manipulator log the
landings. As for logging the
time as pilot time / flight time / PIC time, I'm still not sure I understand how. It seems to me that in the scenario described, nobody gets to log the time, at least until the third landing is completed.
There are more than 30 years of Chief Counsel interpretation letters which point this out, but people still keep asking the question. As Neil is, I think, a lawyer, I would just suggest to him what you did -- read 14 CFR 61.51(e) and see that subparagraph (1)(i) says a pilot rated (or having Sport Pilot privileges) in an aircraft need only be the sole manipulator of the controls, not necessarily acting as PIC, to log PIC time.
And yes, there are about a dozen Chief Counsel letters on this stretching back at least to the Beane letter of 1977.
Yes, I know 61.51(e) well, and am fully aware that you can log PIC without acting as PIC (or even being legal to act as PIC), in certain circumstances. The circumstances that I recall being addressed in Chief Counsel opinions are where either one of the pilots is a CFI or one of the pilots is under the hood. Then, that counts as "flight time" for both pilots, either because the simulated instrument operation has two required crew members or because flight instruction is occurring. Then, as everyone should know by now, a pilot can log pilot time, flight time, and PIC time even if not endorsed, current, etc., so long as the other pilot is a legal PIC.
Here we've got a different scenario. No flight instruction is being given or received, and the operation does not legally require two pilots. The key issue, in my mind, is not 61.51(e) (governing PIC time), but rather 61.1(b)(15) (governing pilot time, of which PIC time is a subset). The non-current pilot (legally a passenger) is not giving or receiving instruction and is not a required crew member, so he does not meet the requirements of 61.1(b)(15). For the purposes of that section, it seems just as if I, as a private pilot, took my non-pilot friend (or even pilot friend who is not rated in category and class) up and let him manipulate the controls. Since I'm not a CFI, he doesn't get to log anything, right? Change the hypothetical to a friend who is an appropriately rated pilot, but not legal to act as PIC (no medical, not current, missing endorsement, no BFR, etc.) and no simulated instrument is involved. How is that any different from the standpoint of 61.1(b)(15)? Certainly nothing in the Beane letter suggests to me that it should be different.