I am solving the problem from the textbook, and g) part states "There is exactly one person whom everybody loves." L(x, y) is "x loves y."
(1) The first and easiest solution is: $\exists !y\forall xL\left(x,y\right)$
(2) Then I came up with the second solution: $\exists \:y\left(\forall \:x\left(L\left(x,\:y\right)\:\wedge \forall \:z\left(L\left(x,\:z\right)\rightarrow \left(z=y\right)\right)\right)\right)$ Which I think is also the correct one.
(3) The third solution is: $\exists x\left(\forall yL\left(y,\:x\right)\:\wedge \forall z\left(\left(z\ne x\right)\rightarrow \exists w\neg L\left(w,z\right)\right)\right)$
(4) And the solution from the book is: $\exists x\left(\forall yL\left(y,x\right)\wedge \forall z\left(\left(\forall wL\left(w,z\right)\right)\rightarrow z=x\right)\right)$
Is the (2) second formula correct in this case? and if so, are they equivalent?
No, it is not correct. $\forall z (L(x,z) \rightarrow (z=y))$ means that if someone loves $z$, that person must be person $y$. Which means that someone can only love person $y$ and no one else. However, "There is exactly one person whom everybody loves." means that there is one person loved by everybody and no other person is loved by everybody. This does not imply that everyone only loves 1 person, which is what you encoded in your formula.
The difference with eq. 4 is that it is now stated that: "if someone is loved by everyone, the person must be person z.". Note that this does not state that everyone only loves 1 person. The forall part is necessary in this case.
A good way to see if you translation is correct is to translate your formula back to English and see how well it matches the original sentence.