Do primitive positive formulas only allow conjunctions, not disjunctions? If so, then injective modules are existentially closed, but are a strictly stronger concept in that the equivalent for infinite conjunctions are allowed.

The concept you're after is that of algebraically compact module (also known as pure inective module). Here's a link to a paper by Prest that talks about some their properties.

Baer's criterion asserts existential closure under infinite conjunctions. Let
$$
(\exists m) r_1 m = m_1 \wedge r_2 m = m_2 \wedge \cdots
$$
be a particular formula, where $R$ is your coefficient ring, $M$ a module over $R$, with $r_i \in R$ and $m_i \in M$. Then consider the ideal $I$ generated by $r_i$ in $R$. There the map that sends $r_i$ to $m_i$ is a homomorphism from $I$ to $M$ if and only if that formula is consistent. (There's a homomorphism if and only if the formula does not imply $s = s'$ for two distinct elements $s, s' \in R$.)