I need to find a complete theory with exactly $3$ non-isomorphic countable models. It is from exercise $2.3.15$ in "Model Theory" (Chang-Keisler 2012).
I think case $1$ in exercise $2.3.15$ is countably saturated because it is $DLO$ without end points. Case $2$ is countably atomic because it is countable, and case $3$ is not countable. But I am not sure about these conclusions and need confirmation.
The first model is not saturated because it does not realize the type $\{x>c_n:n\in\omega\}$.
It is instead atomic. Every type is isolated by one of the formulas $x<c_0$ and $c_n<x<c_{n+1}$ for $n\in\omega$.
The second model is not atomic because it realizes the type $\{x>c_n:n\in\omega\}$ which is not isolated.
Neither it is saturated as it omits the type $\{l>x>c_n:n\in\omega\}$, where $l$ is the limit above.
The third model is saturated. In fact, non-algebraic types either contain $c_n<x<c_{n+1}$ for $n\in\omega$ or they contain $\{x>c_n:n\in\omega\}$. These formulas and type define a DLO. But DLO are $\omega$-saturated.