Having taken a number of pretty cold showers in my time, I can say that holding your head under the frigid spray for a while gives pretty much the same feeling as ice cream headache. Sounds like another vote for blood vessel constriction.
All I can say is that I’ve never experienced ice cream headache. But then, I live a pure and blameless life devoid of excessively indulgent behaviour, so I guess my virtue is its own reward.