The instructor does not have this authority. Assumption of such authority, or anything beyond that detailed in the regulation, is without merit, and without backing. And a little smug.
It is not the instructor's responsibility to ensure the flight is properly logged. As already shown by quoting the regulation verbatim, it is the instructor's responsibility to sign for the instruction given. End of story. Nothing more. That the student may want to write anything of his or her own in that student's own logbook is not truly your concern.
It's not your logbook.
No, it is not. This is about legality, not speculation, not what you wish might be. Do you account "sole manipulator" flight time as the true measure of pilot ability, and the quality of experience a pilot has? Garbage.
I've known a lot of pilots who claim ample hours of "pilot in command time"in various venues, with whom I would never permit my dog to fly. Hours mean exactly nothing. Squat. Zippo.
Quite precisely. Nobody should care about any of the time in your logbook. I know one individual who is flying a 747 for a company, who falsified most of what's in his book, including manufacturing his pilot training and subsequent experience. He knows who he is. He wouldn't be where he is but for his ability to coerce and bamboozle; his ability as a human being and a pilot is poor at best. But his logbook shows the hours.
I can tell enough about a person by the time I taxi to the runway with them, often enough, that I needn't see that person's logbook. The experience one has speaks for itself. Hours are meaningless, and speak nothing to ability, knolwedge, understanding, or judgement. But you go right ahead and log it any which way you like.
