This incident happened during pregnancy but she lasted a while longer before going in. When she did, it was planned and she was there overnight before they decided it was time. But you’re right, “if you care about your baby” could mean it’s urgent. But they were able to wait and if it were life and death I would think they would have prevailed upon her more forcefully.
As it turned out, the cops were able to wake him from his deep sleep and he got there in time. 
With my first, I had been labouring for hours before my water broke with a huge gush. With my second I was just over 36 weeks and had a sort of “off” feeling one morning, so popped in to get checked out to make sure everything was okay. They found I had almost no amniotic fluid left at all, and I was immediately given an emergency induction and an 8 hour time limit before they would have to do a C-section. I had absolutely no idea when or how my water had broken, but I sure was glad I trusted my instinct that day.
Usually the water doesn’t break until a ways into labor so to describe it as the beginning of labor is false. The water breaking does throw the process into overdrive. Labor is best thought of as the process of dilation of the cervix to allow the baby to come out, from zero centimeters to 10 which is fully dialated. Usually active labor is any point in which the cervix has dilated 3 or more centimeters. When inducing labor at about 3 cm if the water has yet to break the doctors will break the sac with a tool. At this point the woman will usually describe the pain as being more intense. After the water breaks there is no turning back and tbe baby is coming out one way or another.
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