Let $M\models T$, a stable theory. Let $a,b\in U\succ M$.
A. Are the following equivalent? (For a comparison, cfr. this other question/answer.)
$\varphi(M,b)\neq\varnothing$ for every $\varphi(x,z)\in L(M)$ such that $\varphi(a,b)$
$\psi(M,M,b)\neq\varnothing$ for every $\psi(x,y,z)\in L$ and $m\in M$ such that $\varphi(a,m,b)$
If I am not mistaken, 2$\Rightarrow$1 is a rephrasing of Exercise 8.3.8 of Tent-Ziegler (weak heir $\Rightarrow$ heir). In the appendix, they give a hint.
B. But, would the following more direct strategy also work? (I hope it is not too sketchy/delusional - in the case, I can expand/delete respectively).
Let $L\big[{\rm ext}(x,y)\big]$ be the language $L$ expanded with a symbol for every formula $\psi(x,y,z)\in L$ where the variables $x,y,z$ are fixed. Let $M\big[{\rm ext}(a,b)\big]$ be the expansion of $M$ that interprets $\psi(x,y,z)$ with $\psi(a,M,b)$.
We also need the intermediate expansions $L\big[{\rm ext}(y)\big]$ and $M\big[{\rm ext}(b)\big]$ which we defined similarly.
Step I. Assume $M$ is saturated and prove $2\Rightarrow1$ as follows. Use stability to conclude that $M\big[{\rm ext}(b)\big]$ is saturated in the language $L\big[{\rm ext}(y)\big]$. Let $m\in M$ be such that $\psi(a,m,b)$. To prove 1 we show that $\psi(m',m,b)$ for some $m'\in M$.
Let $p(y)={\rm tp}(m)$ in the language $L\big[{\rm ext}(y)\big]$. By 2 the type $\{\psi(x,y,b)\}\cup p(y)$ is finitely satisfied in $M$. Hence, by saturation, there is are $m',m''\in M$ such that $\psi(m',m'',b)$ and $m''\equiv m$ in the expansion $L\big[{\rm ext}(y)\big]$. As $M\big[{\rm ext}(b)\big]$ is homogeneous, there is an $f\in{\rm Aut}\big(M\big[{\rm ext}(b)\big]\big)$ such that $f(m'')=m$. Hence $\psi(f(m'),m,b)$ as required.
Step II. Prove that in a stable theory, for every $M,a,b$ and every sufficiently large saturated model $N$ such that $M\big[{\rm ext}(a,b)\big]\equiv N\big[{\rm ext}(a,b)\big]$, where the symbol of elementary equivalence refers to the language $L\big[{\rm ext}(x,y)\big]$. Note that both 1 and 2 hold for $M$ iff they hold for $N$.