Definition Problem for toy SPN
Definition 3.1 : Let $\pi_S:\{0,1\}^m\rightarrow \{0,1\}^n$ be a S-box. Consider an (ordered) pair of bitstrings of length $m$, say $(x,x^*)$. We say that input XOR  of the S-box is $x\oplus x^*$ and the output XOR is $\pi_S(x)\oplus\pi_S(x^*)$.
Now, the point of confusion is how can the output of the S-box be separated as $\pi_S(x)\oplus\pi_S(x^*)$ when the input is $x\oplus x^*$?

The crux of your question is above. Differential cryptanalysis is a chosen plaintext attack.
The attacker presents the Sbox (more generally the full cipher) with two sets of chosen plaintexts with a chosen difference $d$
$\{u_i: 1\leq i \leq N\},\quad \{u_i\oplus d: 1\leq i \leq N\}$
and computes the set of output distributions corresponding to this input difference
$\{\pi_S(u_i)\oplus \pi_S(u_i\oplus d):1\leq i \leq N\}.$