I think this is backwards. I found this discussion of the ZIP parameteres helpful.
Theta is the usual rate parameter of a poisson distribution. You can confirm this by direct comparison of the 2nd branch of the ZIP PDF shown on the page you linked, and the PDF of a normal Poisson (which you recover when phi = 1).
So you want to estimate theta as usual in a normal Poisson setup: do some stuff to get logits, then set theta = exp(logits).
Psi is related to how many “extra” zeros you want to generate. As noted above, when phi = 1, you get a standard Poisson PDF, so all the zeros in your data are assumed to come from a Poisson. When phi = 0, you are saying that your Poisson process NEVER ONLY generates zeros.
So you could set psi ~ Beta(1, 1) if you want to be uninformative, or you could try to look at your data and try to skew it one way or the other, but you’ll probably want to use some kind of beta prior.