= Solution
Write $\lambda=\sigma^2$, $v=\sigma_0^2$, and $a=\sqrt\rho$. We keep the model's printed coefficient $\sigma^2Z_i$ literally. Its latent <log odds> can be written
$$
\theta_i=x_i^T\beta+\lambda Z_i+\eta_i,\qquad\eta_i\overset{\mathrm{iid}}\sim N(0,v).
$$
The independence of the $\theta_i$ is conditional on $Z$: they are not marginally independent when the <random effects> $Z_i$ are correlated.
Holding the other predictors and the random-effect realization fixed, increasing predictor $j$ by one shifts the latent <log odds> by $\beta_j$, so it multiplies the conditional odds of a late departure by $e^{\beta_j}$. For a binary snow indicator, this compares the snow and no-snow days with other predictors held fixed. This is a conditional <odds ratio>, not generally the same as a marginal odds ratio after integrating the random effects.
Randomizing $\theta_i$ allows unmeasured daily conditions to change the success probability beyond the systematic <linear predictor>. In this <logistic-normal regression with autoregressive random effects>, let $P_i=\operatorname{logit}^{-1}(\theta_i)$ and $\bar p_i=\mathbb E[P_i]$. The <law of total variance> gives
$$
\operatorname{Var}(Y_i)=n_i\bar p_i(1-\bar p_i)+n_i(n_i-1)\operatorname{Var}(P_i),
$$
so the mixture allows <overdispersion> beyond a <binomial distribution> with fixed probability. The correlated part also allows related outcomes on nearby days.
The stationary <autoregressive process of order one> has $\operatorname{Var}(Z_i)=1$ and $\operatorname{Cov}(Z_i,Z_j)=a^{|i-j|}$. Hence
$$
\boxed{\operatorname{Var}(\theta_i)=\sigma^4+\sigma_0^2,\qquad
\operatorname{Cov}(\theta_i,\theta_j)=\sigma^4\rho^{|i-j|/2}\quad(i\ne j).}
$$
The $\sigma_0^2$ term is unstructured daily variability, whereas $\sigma^4$ is the temporally correlated component under the printed parameterization. Larger $\rho$ gives greater persistence: the lag-$h$ correlation of the latent log odds is $\sigma^4\rho^{h/2}/(\sigma^4+\sigma_0^2)$.
The PDF itself prints both the coefficient $\sigma^2Z_i$ and the requested combination $\sigma^2+\sigma_0^2$. They are inconsistent as a marginal variance. If the intended coefficient were $\sigma Z_i$, the variance and off-diagonal covariance would instead be $\sigma^2+\sigma_0^2$ and $\sigma^2\rho^{|i-j|/2}$. The prior's argument list also repeats $\sigma^2$; its displayed density indicates the intended second variance parameter is $\sigma_0^2$.
There is a further substantive issue with that <improper prior>. After integrating the latent variables, the observed-data <likelihood function> is positive and continuous at $\lambda=0$ for fixed finite $\beta$, positive $v$, and $\rho\in(0,1)$. On a compact positive-volume set of those parameters it therefore has a positive lower bound for small $\lambda$. The prior density is $1/(\lambda v)$, so integrating over $\lambda$ gives $\int_0^\varepsilon d\lambda/\lambda=\infty$. Thus the stated joint <posterior distribution> is improper, an instance of <improper posterior from a log-uniform random-effect scale prior>. The requested fixed-parameter conditional laws below are nevertheless proper; their existence does not remedy the joint impropriety.
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