It is correct to say that a connected graph is only when there exist some vertex that is connected to all other vertices?
I think this is correct, because a connected graph not all vertices are connected together, but at least no vertex is not connected? Isn't that the above definition is saying?
I know the standard definition but I want to know if this definition is also correct. Thanks
That is correct. A graph is connected if and only if for all $x, y \in V(G)$, there exists a path from $x$ to $y$.
When talking about directed graphs, we have the concepts of weak connectivity vs. strong connectivity. In a weakly connected graph, we are guaranteed either a directed $x-y$ path or a directed $y-x$ path, but not necessarily both. A strongly connected path guarantees us both a directed $x-y$ path and a directed $y-x$ path.