On the definition of locally connected spaces I have seen two a priori different definitions of locally connected space : 

1) For all point $x$, and neighborhood $V$ of $x$, there is a connected neighborhood $C$ of $x$ such that $C\subseteq V$

2) For all point $x$, and open set $U$ containing $x$, there is an open connected neighborhood $O$ of $x$ such that $O\subseteq U$.

I imagine those two definitions are identical, but I don't see why.

The same thing for locally path-connected spaces, we can either use open neighborhoods or just neighborhoods, are the two definitions the same?
By definition a neighborhood of a point $x$ is a set $I$ that contains an open set $A$ such that $x\in A$. So 1) $\Rightarrow$ 2) by considering $U$ as your neighborhood; 2) $\Rightarrow$ 1) by considering the open set of the definition of neighborhood.
By definition a neighborhood of a point $x$ is a set $I$ that contains an open set $A$ such that $x\in A$. So 1) $\Rightarrow$ 2) by considering $U$ as your neighborhood; 2) $\Rightarrow$ 1) by considering the open set of the definition of neighborhood.