question about two different definitions of limit point From the definition of a limit point it follows  :

A limit point $x$ of a subset $S$ of a topological space $X$ is a point such that every neighborhood of $y$ has a point in common with $S$.

also based on Wikipedia :

it doesn't make a difference if we restrict the condition to open neighborhoods only.

I know that if every neighborhood of $y$ intersects $S$, then also open neighborhoods of $y$ has a point in common with $H$
but why we can claim that if every open neighborhood of $y$ intersects $S$ then, so does every neighborhood of $y$?
First, by your question, you seem to have understood this direction but I will write it anyways: open neighborhoods are a special case of neighborhoods. So, if every neighborhood of $y$ intersects $S$, the same can be said for every open neighborhood.

Now, for the other direction, assume every open neighborhood of $y$ intersects $S$ and let $V$ be a neighborhood of $y$. By definition, $V$ contains an open set $U$ which contains $y$. $U$ is an open neighborhood of $y$, so $U$ intersects $S$. But $U\subset V$, so $V$ also intersects $S$. $V$ is an arbitrary neighborhood, so every neighborhood intersects $S$.
Mathematicians often shorten the statement 


  $U$ is an open set containing $x$


to the phrase 


  $U$ is a neighborhood of $x$.


Borrowing from Munkres: 


  If $A$ is a subset of the topological space $X$ and if $x$ is a point
  of $X$, we say that $x$ is a limit point (or “cluster point”, or
  “point of accumulation”) of $A$ if every neighborhood of $x$
  intersects $A$ in some point other than $x$ itself. Said
  differently, $x$ is a limit point of $A$ if it belongs to the closure
  of $A − \{x\}$. The point $x$ may lie in $A$ or not; for this
  definition it does not matter.