I'm confused about the definition of a Euler cycle, my book says "a cycle in a graph G that includes all of the edges and all of the vertices of G is called an Euler cycle"

However on wikipedia it says "An Eulerian cycle, in an undirected graph is a cycle that uses each edge exactly once."

Which one is it?

If it is the latter, then is it possible to travel all the edges once and touch every vertex (not necessarily once) given the graph G has a Euler cycle? If so, how do you prove it?

Many thanks!