My first 2 children were overdue by about 10 days.
I went into labour naturally with my first at 9pm and she was delivered at 5.30pm the next day by vac suction. I think because I went private with my 2nd baby my OB didn't want me to go too long overdue (despite me having a history of it) so I was induced at 8am and delivered vaginally just after midnight that night. I wish I had been left alone to go into labour naturally. Neither of my over due babies had any signs of being 'over cooked' and I had good amniotic fluid levels and the placentas were good too. I think my body just needs a bit longer than other women...

My third baby was induced because I had cholestasis at 37 weeks but I bet he would have been overdue as well.