How is the counting-rate of particles obtained in an experiment?
It depends on the experimental setup.
Experiments measure neutrino oscillations by starting with a known neutrino beam, measuring its content in the three standard model neutrinos, i.e. have detectors that count how many there are at one specific location, then put a detector far enough away (depending on the energy of the beam) and count again the percentages of the standard model neutrinos. Two locations give the purely in count rates . Discrepancies between expected by the standard model counts and counts found by the experiments have given rise to the sterile neutrinos proposals.
how do we know what we are counting if not by knowing the energy and distance-dependent characteristic
There are only two locations on the beam path at which the number and type of neutrinos are counted. The energy is not recorded, because the neutrinos as weakly interacting neutral particles need very specific detectors just to count muons, electrons, and (more difficult) taus, by their charged interactions. Just the number of the standard model neutrinos, checked by the detection of the number of electrons or muons or tau at the end detector.The hypothesis of extra neutrinos to which the usual three can oscillate needs new experiments with discrimination in the energy of the end events.