When to let them induce

Anon
I'll be 40 weeks and due on Thursday, June 15th and also have an appt that day for an ultrasound and then an appt. The next day to do a follow up and schedule for a day to induce if she isn't here yet. Doctor said last week something about inducing at the week of being 41 weeks, I really would like to be induced while my bfs mom is in Europe (less visitors and stress with family to worry about) I don't plan on having anyone come the first day she born bc even if I get induced IN I'M REALLY COMMITED to all natural no meds or anything buttt I'm scared bc most people who get induced end in c section. So after the 15th what do you think would be a good time to schedule the induction for ? Also if anybody has any stories on being induced and still having your baby naturally or with epidural that didn't end in c section id love to hear it for comfort. This is my first baby ❤