What is the correct definition of "étale cover" In many books, an étale covering (used for defining étale fundamental group) of a scheme $X$ is defined as $f:Y\rightarrow X$, where $f$ is a finite étale morphism.
In some other books, $f$ should be a surjective finite étale morphisms.
Why do we have two different definitions? 
 A: The better definition is the one that does not require that an étale cover be surjective.
Here are some reasons:   
1) The original definition in SGA 1 (Définition 4.9,page 5) by Grothendieck and his students does not require étale covers to be surjective.
2) Occam's razor: do not introduce extraneous conditions which don't seem to serve any purpose.
3) Étale covers in algebraic geometry are to be analogues of finite covers in topology.
But coverings in topology shouldn't be assumed surjective : this is for example  Bourbaki's point of view in his new volume on Algebraic Topology (see page 68).
     4) Here at the end of page 2 is Chambert-Loir's argument, expressed in the most elevated form of French poetry, the alexandrine.
The lamentably prosaic rendition of his argument is that requiring surjectivity would destroy the category equivalence between étale covers and  $\pi_1$-sets.
Chambert-Loir diplomatically describes the incorporation of surjectivity in the definition of étale covers as "étrange", which means "strange".
