Let's say we have the system of $l$ equations below: $$ \left\{ \begin{array}{c} a_{11}^{a_{12}}=a_{13} \\ a_{21}^{a_{22}}=a_{23} \\ \ldots \\ a_{l1}^{a_{l2}}=a_{l3} \\ \end{array} \right. $$
Here, every $a_{ij}$ is actually a variable chosen from a set of variables $V_{I}=\{b_1 , b_2 , \ldots b_n\}$, such that the three corresponding variables(the b's) in each equation are distinct and each $b_i$ occurs in the system. Denote this system with I and assume that there exists a set of n different integers(one for each $b_i$) which is a solution.
Is it always true that the same system(but with powers changed to additions everywhere) denoted with II: $$ \left\{ \begin{array}{c} a_{11} + a_{12}=a_{13} \\ a_{21} + a_{22}=a_{23} \\ \ldots \\ a_{l1} + a_{l2}=a_{l3} \\ \end{array} \right. $$
must also have a solution - a set of n distinct integers? Here, each $a_{ij}$ refers to the same variable from the set V as in system I. [The example in the comment should clarify the set up]
Note 1: The answer seems to be 'YES', since taking powers is much more complicated than doing additions). How can we prove it?
Note 2: We would have a trivial counterexample if we include the two equations $x^y =p$ and $y^x = q$ in I. Thus we must also add the constraint $\{a_{i1},a_{i2}\}\neq \{a_{j1},a_{j2}\}$, for all $i\neq j$.
Since the goalposts keep moving, I find myself here again. And because I don't ever want to have to come back, rather than spending time answering the question you are currently asking, I will reformulate your current question and ask you whether it captures what you actually mean to ask.
Let $I_\ell$ and $II_\ell$ be the two frames you specify, each containing $\ell$ equation. (I note that you do not require equations to be distinct. Requiring or not requiring distinctness does not change what follows.)
Let $R$ be the set of "numbers". Let $\mathscr{P}$ denote the powerset function and let $\phi$ be the (family of) forgetful functor(-s) $f:R^n \rightarrow \mathscr{P}(R): (r_1, r_2, \dots, r_n) \mapsto \{r_1, r_2, \dots, r_n\}$. (Note that $\phi$ forgets duplicates in the $n$-vector.) Define a solution pool, $C_k$, to be an element of $R^k$ where $|\phi(C_k)| = k$.
Define the cardinality spectrum of a frame $F$, containing the variables $(a_{ij} \mid 1 \leq i \leq \ell, j \in \{1,2,3\})$ to be the set of $k$ such that there exists a solution pool $C_k$ where $F$ has a solution with $a_{ij} \in C_k$ for $1 \leq i \leq \ell, j \in \{1,2,3\}$.
The Question asks: Is the cardinality spectrum of $I_\ell$ always a subset of the cardinality spectrum of $II_\ell$.
Does this capture what you actually want to know?
Prior Answer
Let $a_{i1} = a_{i3} = i$ and $a_{i2} = 1$. Then $S_I = [1,l] \subset \Bbb{Z}$. The $i^\text{th}$ equation in II requires $\{i,1,i+1\}$ which are all in $S_I$ except for the $l^\text{th}$, which requires the new integer $l+1 \not \in S_I$. Consequently, with this choice of I, $S_{II} = [1,l+1]$ has larger cardinality than $S_I$.