When passing a waypoint, when exactly are we on the next leg?
The transition point as defined in RTCA DO-283 RNP MOPS is the point where the aircraft's path passes through a vertical plane that bisects the inbound and outbound legs going through the waypoint.  

That generic description covers all the possibilities of the aircraft passing the waypoint regardless of how good or bad it is following the defined path.  

Realistically, in the case of the fly-by waypoint, it's the point half way through the transition turn where the waypoint is directly abeam the flight path.  

The same rule (crossing the bisecting plane) applied to the fly-over waypoint results in the transition occurring at the waypoint.