Use quantifiers to express: "There is exactly one person whom everybody loves"

Question

If $L(x, y)$: “$x$ loves $y$”, the domain consists of all people in the world.

Use quantifiers to express: "There is exactly one person whom everybody loves".

1. $\exists x(\forall y L(y, x) \wedge \forall z(\forall w L(w, z) \implies z = x))$
2. $\exists x \forall y( L(y,x) \wedge \forall z( L(y,z) \implies z = x ))$

So, 1 is standard answer, 2 is my answer, is my answer correct?

Answer 1

No, (2) is much stronger than (1) because (2) requires that everybody (i.e., $y$) loves only $x$ and nobody else.