I think you are on the right track with that documentation link, I would recommend you re-read that example very closely. To summarize what is in the documentation, the binomial cluster model you are fitting is a two-component mixture, where the first component is binomial with 'n' trials and success probability 'mu_star + mu', the second component is binomial with 'n' trials and success probability 'mu_star', and the mixing probabilities are represented by pi and (1 - pi). Furthermore, mu_star = (1 - mu)*pi (where pi is still the mixing probability).
They show in the linked example that the estimate for the mu parameter, 'mu_hat', is computed as the inverse link of the intercept parameter in the model for mu (specified via the model statement). In the documentation example, mu_hat is equal to 0.5831. Likewise, in Table 43.11, they show how to compute estimates for the mixing parameter ('pi_hat') based on the coefficients from the model specified in the probmodel statement.
I believe that the pred statement for this model is generating the success probabilities for each of the two components of the mixture model, i.e., pred1 is mu_star_hat + mu_hat, and pred2 is mu_star hat. So, for example, for PHT = 0 and TCPO = 0, pi_hat = 0.6546, therefore mu_star_hat is (1 - 0.5831)*0.6546 = 0.273 (rounded), and mu_star_hat + mu_hat = 0.5831 + 0.273 = 0.865. Those are the values of pred_2 and pred_1 that I get, respectively, when PHT = 0 and TCPO = 0.

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