what makes the Foundation special enough that only their members reincarnate
Technically, that would be "what makes the Foundation special enough that only their members reincarnate with memories of previous lives intact."
'why do such different traumas produce the same stereotyped set of reactions'. It sort of ends up implicitly suggesting a unified theory of anomalies, which is a suggestion I never really like.
This certainly has no attempt towards the unified theory. The instances are effectively having PTSD, and the second behaviour relates to it. Perhaps the first behaviour (to not be in the light) is more unusually uniform. But I would argue that it was the darkness and loss of consciousness that relieved them of whatever suffering they went through prior to death, and that's really the one universal factor among almost all instances.
I don't buy that only Foundation people would come back, or would be traumatised so severely by those deaths that they're a special case.
Upon reflection, while the article does have the Foundation assume that the instances to be former Foundation personnel, the instances never explicitly declared that they are Foundation personnel (not to mention that not all instances are shown in the article). And besides, only one of the scenarios listed here can be tied directly to the Foundation (the nuke scenario). The rest can be decontextualised such that the person dying need not be Foundation personnel.
But whether the person is former Foundation member or not does not matter, for any instance might have important information regarding anomalies.
EDIT: In fact, the 1st and last scenario makes more sense if it is the Broken Church culling a heretical Ticker, instead of a Foundation member being ceremonially executed in a cruel and interesting way.