I think if you asked most astrologers the answer would be the moment when the child takes his/her first breath. I'm not sure whether they would have reflected on the cutting of the cord, but if they have, then I think they would still come down to that first breath.
I suppose one could argue that the key property is the moment when the child becomes a viable independent organism, which might lead to a debate about whether then child is viable if still attached through the cord. My feeling would be yes, though a failure to cut the cord (possibly because of the incapacity of the mother and the lack of medical attendants) might lead to real complications.
A perhaps more important question is Does it matter how we define birth?
The answer suggested at the outset is that it does matter for some approaches to Astrology - for example the use of Primary Directions is usually said to require an accurate time because an error of 1 degree in the Ascendant (on average 4 minutes of time) could through out predictions by a year. This has led Astrologers to seek to 'rectify' natal charts - 'rectify' of course means 'to put right' which strongly implies that there is a 'right time of birth'
Like zhan.thay, I have a doubtful birth time, and I've played with rectification, mostly because I like playing with software, rather than a real need to find my 'true' birth time. From what I can see there are a myriad of methods, and even a £300 program 'JigSaw' which seek to help you do this. Most of the more recent methods attempt to fit events in your life to a prediction method, such as PDs or Secondary Progressions, to come up with a 'best fit' - note not the 'correct' answer. Improving on the forecast accuracy for one event may well reduce the forecast accuracy for another and overall I tend not to bother with such an approach.
An interesting book, 'Moment of Astrology' by Geoffrey Cornelius questions the need and indeed basis for seeing an event being linked to one unique moment and indeed argues that valid charts, some of which were cast in error, can still be found which provide a useful but different insight into the person, or event being considered. He illustrates the natal side of this by looking at the intial birth time given for Princess Diana and then the later 'corrected' version and argues both charts were valid in telling us something about her.
So it might be interesting to cast a chart for Winterchild's children based on first breath and then the later cord separation and see not which was right but what additional information one gives compared to the other that can be verified from later life.
Incidently, I know from experience that a wrongly cast horary chart (usually because I forgot to allow for daylight saving) can still 'answer the question'.