I agree completely with your ObG doc. Transmission of warts to babies is uncommon, and generally occurs from moms who have overt, visible warts. Even then, most providers do not consider it an indication for cesarean delivery. Since your warts cleared up 2 years previously, it is very unlikely you were infectious at the time of delivery.
As I have said many times on this forum, there is uncertainty whether people who have had HPV continue to carry the virus and for how long. It is possible that viral DNA often (always?) persists in tissues for life; the experts disagree on this. However, in the vast majority of cases, even if DNA persists, the body's immune system keeps the virus in check, they never have another recurrence of warts or other abnormality, and never transmit the virus either sexually or during delivery. So for practical purposes, they are cured.
Bottom line: Probably you are home free, will never have warts again, and your baby probably was not infected. To be super safe, make sure the pediatrician knows your history; s/he can then keep a lookout. But I really believe you have no worries.
Good luck-- HHH, MD