For $L$ structures $A$ and $B$ we write $A\preceq_{1}B$ if $A\subseteq{B}$ and $A\models{\varphi(a)}$ iff $B\models{\varphi(a)}$ for any finite tuple (of the correct length) $a$ from $A$ and for any universal formula $\varphi(x)$.
An $L$ theory $T$ is 1-model complete if whenever $A\preceq_{1}B$ with $A,B\models{T}$, we have $A\preceq{B}$.
There are several equivalent statements for a theory being model complete (see here: http://modeltheory.wikia.com/wiki/Model_completeness for a list of equivalent statments). Are there such equivalence statements in the above case. And if so what would be a good reference for them?
You might find what you're looking for in this paper.