But, to be frank, any flight where it's not absolutely clear who the captain is, before the aeroplane moves under its own power, is doomed at the very least to suffer confusion, and very much at risk of accident, before it starts.
Very true. When I "check fly" I state that the "other" pilot is PIC, and I will brief, then intercede, only as required, and that won't relieve them of PIC duties. I'm trying to push the duty to them, not retain it for myself. I have flown more than 50 hours of "check" or mentor flying this year, and there's not a minute of it in a log book anywhere, the "candidate pilot" logged it. I want
them to have the experience, that's why I'm flying with them!
I rarely "sign for" an aircraft, though when I took helicopter training I did. 'Signed that I took it (and responsibility if it, and how it was flown) and that I returned it X.X hours later.