The barbers paradox first order logic formalization

Question

I tried to look on the site and while I found some similar questions, I did not find the first order logic formalization of the following sentence (the basic barber's paradox), so I wanted to ask if I got the right first order logic formalization of it, and I am sure others will benefit from this thread as well.

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

My attempt: this complicated sentence can be made simpler by inferring that "for all of those who does not shave themselves, are shaved by the barber"

And now it seems to be easier to substitute with variables so:

$$\exists x(\lnot S(x,x) \rightarrow S(b,x))$$

where $S(x,x)$ is shaving(verb), $x$ are the persons (who don't shave themselves), and $b$ is a shortcut for barber, so if a person does not shave himself, it can be inferred that he is shaved by the barber.