Petr Pudlák's answer is of course exactly right about the equivalence, and he gives a pair of proofs which shows why it holds. But it is worth remarking as a footnote that this equivalence is (of course!) only available if you are already using the language of first-order logic with identity.
Now, it is natural and indeed pretty standard first to (1) introduce the language of first-order logic first without identity and then (a later chapter!) (2) add the identity relation. Note, then, that "Everyone loves himself or herself" can already be perfectly well rendered into our formalism at stage (1): we do not have to have identity explicitly in the formal language do to the translation.
It is a good principle, when rendering English into the language of FOL, to only expose as much structure as we need (translating as simply as we can, without unnecessarily going round the houses -- as with any translation). That is why the simpler translation already available at level (1) would be preferred as a translation to the unnecessary circumlocution of the more complex (though provably equivalent) sentence that becomes available at level (2).