How to write down a 3-level hierarchical model with a level-1 equation (reduced form)? Suppose longitudinal data on $y_{rct}$: the average temperature of region $r$ in country $c$ at time $t$. You $y_{rct}$ fit a 3-level hierarchical linear model with country-year and country deviations from an overall mean. Assuming $y_{rct}$ is centered at 0, a model could be (variance hyper-priors omitted):
$$y_{rct} \sim N(\mu_{ct}, \sigma_{rct}) \\
\mu_{ct} \sim (\mu_c,\sigma_{ct})\\
\mu_c \sim N(0,1)\\$$
This notation makes clear that the mean at level 1 is a function of the mean at level 2, which is a function of the mean at level 3. However, in popular software like the lme4 R package or Stata, the code to produce a 3-level model usually involves a single-level model with independent country and country-year offsets to the grand mean. For example, here or p.7 here. This seems to be equivalent to something like:
$$y_{rct} \sim N(\mu_{ct} + \mu_c, \sigma_{rct})$$
QUESTION
While Model 2 adds the same deviations to the mean of $y_{rct}$, are these models actually equivalent? My hunch is no: Model 1 shrinks the $\mu_{ct}$ toward its respective $\mu_c$ but Model 2 shrinks $\mu_{ct}$ toward the grand mean of 0. However, perhaps I am missing something about how these work that makes Model 2 the only way to specify nested intercepts in most software.

Your question, "While Model 2 adds the same deviations to the mean of $y_{rct}$, are these models actually equivalent?" isn't the same as the question title, "How to write down a 3-level hierarchical model with a level-1 equation (reduced form)?". @statmerkur has answered the question title, but I'm just going to answer the former question.

$$y_{rct} \sim N(\mu_{ct}, \sigma_{rct}) \\
\mu_{ct} \sim (\mu_c,\sigma_{ct})\\
\mu_c \sim N(0,1)\\$$
is not quite equivalent to
$$
y_{rct} \sim N(\mu_{ct} + \mu_c, \sigma_{rct})
$$
as the $\mu_{ct}$ terms do not mean the same thing in both cases.
However, your first equation is equivalent to
$$y_{rct} \sim N(\mu_c + \gamma_{ct}, \sigma_{rct}) \\
\gamma_{ct} \sim N(0, \sigma_{ct})\\
\mu_c \sim N(0, 1)\\$$
where $\gamma_{ct} = \mu_{ct} - \mu_c$ (e.g. how much country $c$ at time $t$ deviates from the overall mean for country $c$). It follows that shrinking $\gamma_{ct}$ towards $0$ is equivalent to shrinking $\mu_{ct}$ towards $\mu_c$.
