= Solution
An <atomic model> is a model in which every finite tuple realizes an isolated <complete type> over the empty set. For a <complete theory> in a <countable> language, every <countable> atomic model is prime. The underlying argument is a <back-and-forth method> in the one-sided form needed for an embedding: enumerate the atomic model, and extend a finite partial elementary map one element at a time using an isolating formula for the enlarged tuple. The target model has the required witness because the already matched tuple has the same <complete type>.
Thus property (iii) implies property (iv). It need not imply pointwise definability or term-generation. The <countable> <dense linear order without endpoints> with a single named point, for example $(\mathbb Q,<,0)$, is atomic: <quantifier elimination for dense linear orders without endpoints> isolates a tuple by its finite order-and-equality pattern relative to zero. But order <structure automorphisms> fixing zero move every nonzero rational. Such elements are not parameter-free definable, and no <closed term> names them. This supplies the failed implications (iii)$\not\Rightarrow$(ii) and (iii)$\not\Rightarrow$(i).
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