Why the given statement can't be expressed using predicates and quantifiers in the way described in details?
Rephrasing that second version back into English:


  There exists a person $x$, such that if $x$ is a student, then $x$ has gone to Mexico.


If there exists anyone who's not a student, this ends up being true. "If false then [anything]" is always true, so when "$x$ is a student" is false, "if $x$ is a student, then $x$ has gone to Mexico" is true—whether or not $x$ has gone to Mexico.

What you want is:


  There exists a person $x$, such that $x$ is a student, and $x$ has gone to Mexico.


That is indeed what the first given solution means.