Could you prompt me, example of complete theory, that has, say, two models, $\mathfrak{A}$ and $\mathfrak{B}$, that $\mathfrak{A} \le_{end} \mathfrak{B}$, but $\mathfrak{A} \npreceq \mathfrak{B}$?
For example, for this reason I leave considering of arithmetic vector spaces, with different dimensions, say, R^3 and R^7 - R^7 is elementary extension of R^3, so it isn't fit.
Let $L = \{\leq\}$, and let $T$ be the complete theory of $N = (\mathbb{N}^*,\leq)$, the reverse order on the natural numbers. Let $M = ((\mathbb{N}\cup \{-1\})^*,\leq)$, so $M$ adds a new maximal element $-1$ on top of $N$.
Then $M$ is an end extension of $N$, and it is also a model of $T$ since it is isomorphic to $N$, but it is not an elementary extension: let $\varphi(x)$ be the formula $\forall y(y\leq x)$ which expresses that $x$ is a maximal element. Then $N\models \varphi(0)$ but $M\not\models \varphi(0)$.