Understanding Fourier Coefficient
You have a typo, $x$ is also in $\mathbb{F}_2^4,$ since the Present Sbox is $4\times 4$ bits.

The quantities $S(x),a,b,x$ all lie in $\mathbb{F}_2^4,$ and the inner products such as $\langle b,S(x)\rangle$ etc lie in $\mathbb{F}_2.$ We now treat the inner product values as in $\mathbb{N},$ so we can calculate
$(-1)^{\langle b,S(x)\rangle+\langle a,x\rangle}.$

Basically if the two inner products in the exponent match we have a $+1$ in the overall sum, otherwise a $-1$. Thus the sum computes the imbalance function defined as $$2^4-2d_H(\langle b,S(x)\rangle,\langle a,x\rangle)$$
between the two quantities in the exponent as $x$ ranges over $\mathbb{F}_2^4.$

Here $d_H$ is the hamming distance along the truth tables.