The barbers paradox first order logic formalization
You're not being asked to do any inference; just to express something.


  there exists a barber who shaves all the people that don't shave themselves


translates directly as

$$\exists b\, \forall x\,\big(\neg S(x,x) \rightarrow S(b,x)\big)\,.$$

(There exists a barber such that everyone who doesn't shave themself is shaved by the barber.)

Your version says that there exists a person who, if they don't shave themself, is shaved by the barber. That's not equivalent. For example, it's true in any world where at least one person does shave themself: just let $x$ be that person, so $\neg S(x,x)$ is false, so $\neg S(x,x)\rightarrow\text{anything}$ is true.