NP languages definition
Is it good to define a language $\mathcal{L}$ in NP as a language for
  which, given an element $x$, it is possible in polynomial-time to
  check whether $x \in \mathcal{L}$ or not?


No. If you could do that you could "check" that this language would be in P, because you can check (or more formally decide) all possible words for membership in polynomial time this way.

A correct formulation would be:


  A language $\mathcal L$ is in NP if for every word $x$ in the language there exists a witness $w$ which is of length polynomial in the length of $x$ and given the witness and the word you
  can verify language-membership in polynomial time.


Note that you only need said witness (sometimes also called certificate) need only exist for words that actually are in the language, i.e. you don't need to be able to construct them for negative instances and constructing them is allowed to take super-polynomial time.