= Solution
An element $a$ is parameter-free definable if some formula $\theta(x)$ of the original language has $a$ as its unique solution in the structure. Saying this for every element makes the model <pointwise definable>.
If a finite tuple has individually defining formulas, their conjunction uniquely defines that tuple. Its <complete type> is isolated by this conjunction: for each formula $\varphi$, the sentence saying the unique tuple does or does not satisfy $\varphi$ is decided by the <complete theory>. Thus
$$
\boxed{\text{pointwise definable}\ \Longrightarrow\ \text{atomic}.}
$$
It follows that the <countable> model is also prime. The converse implication to property (i) fails. In $(\mathbb N,<,0)$ every element is defined by having exactly $n$ predecessors, but the only <closed term> in this language is the constant $0$. This infinite model is pointwise definable and is not term-generated.
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