Is it correct to say that 400 ft is the minimum height for start of third segment because ..
Can't recall ever having read the definitive history. However, it will be found in some dusty ICAO tome of the 50s which led to its embodiment in the Design Standards.
the sooner you get the aircraft cleaned up, the better it will climb single engine (two engine types) and generally better obstacle clearance.
.. but don't hit the ground whilst doing so ...
On the other hand, it is now common for manufacturers to recommend 400 ft as the minimum "safe" height to be attained before actioning serious check list items
I've always presumed that this development was solely from considerations of standardisation .. ie concentrate on flying intially to get away from the ground and, as some appropriate level, the presumption is that there will be a more sensible time to worry about the side issues ...