 Solution

A 1-embedding, or existential embedding, is a structure embedding $\alpha :A\to B$ satisfyingfor every quantifier-free $\theta$ and every parameter tuple $\mathbf{a}$ from $A$. The forward direction is automatic for an embedding; reflection of existential formulas is the extra property. For an inclusion this says that $A$ is simply closed in $B$. This is weaker than requiring an elementary embedding, which preserves all formulas.