he/she doesn't own that FI
No but as they have paid for the
FI's time for that flight then don't you think that they deserve a signature to say the flight was indeed carried out with that FI and that the comments entered are correct? I certainly wouldn't log anything as PUT without a signature to say that the training had indeed been carried out. If they don't sign then I assume I am PIC right?
I think it should be a requirement that all FI time is signed by the instructor, to avoid any possible confusion later on. Especially if the flight is for the purpose of the issue of or revalidation of a rating - which it could be ?