After reading the description of the difference between endpoint independent and address restricted, it seems as though what must be happening is that the DNS responses I am receiving must be coming from a different IP address than the one I made the connection on (as endpoint independent allows that, whereas address restricted does not). It seems reasonable there could be some kind of load balancing going on at OpenDNS that farms queries directed at the main OpenDNS IP to a stable of responders that live at different IP addresses.
Which also suggests that maybe if I turned DNS relay back on and left the NAT settings at endpoint independent, everything would work OK as well (it definitely did not with the NAT settings set to address restricted, at least not consistently, sometimes welcome.opendns.com told me I was not using OpenDNS, sometimes it told me I was).
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bc