Suppose that there are two sampling distributions that satisfy (sorry about the lousy math notation) the proportionality relationship,

Pr1(data | parameter) = k * Pr2(data | parameter)

where k may depend on the data but not on the parameter. Then the same proportionality relationship holds for the prior predictive distributions,

Pr1(data) = Integral { Pr1(data | parameter) * Pr(parameter) d(parameter) }
Pr1(data) = Integral { k * Pr2(data | parameter) * Pr(parameter) d(parameter) }
Pr1(data) = k * Integral { Pr2(data | parameter) * Pr(parameter) d(parameter) }
Pr1(data) = k * Pr2(data)

Now write out Bayes' theorem:

Pr(parameter | data) = Pr(parameter) * Pr1(data | parameter) / Pr1(data)
Pr(parameter | data) = Pr(parameter) * k * Pr2(data | parameter) / (k * Pr2(data) )
Pr(parameter | data) = Pr(parameter) * Pr2(data | parameter) / Pr2(data))

So it doesn't matter whether the data were sampled according to Pr1 or Pr2. You can check that the binomial and negative binomial distributions satisfy the proportionality condition by looking them up in Wikipedia.