You make an excellent point. It is important to clarify exactly what practitioners mean when they tell you their cesarean rate. For care providers who do not perform cesareans (all midwives and most family docs), their cesarean rate conventionally means the rate in women still eligible for that practitioner's care at the onset of labor. So a woman who goes into preterm labor would not be eligible for an out-of-hospital midwife's care. And for a home birth midwife who did not have hospital privileges, the woman would have transferred out of her care if she was admitted to the hospital for induction--I specified hospital admission for induction because home birth midwives may use various induction techniques that don't involve hospital admission. But for midwives and family docs with hospital privileges, the woman having labor induced would usually still be cared for by her midwife or family doc, that is, unless the reason for induction was a medical complication outside of their scope of practice.  

-- Henci