First, for the sphere in above picture, isn't the metric supposed to have $sin^2\theta$ in front of $d\phi^2$? I just need to check
And for the main question, what does it mean by the three manifolds are same as differentiable manifolds? The given maps between them seem like diffeomorphisms, but what do the maps have to do with the metric? Could anyone please explain?
The examples I, II, III are all diffeomorphic to each other; that's what "the same" means in the context of diffeomorphic manifolds.
And the point of these examples is that the maps do NOT have anything to do with the metric: a diffeomorphism ignores the metric. In order to demonstrate that I, II, and III are all diffeomorphic to each other, you need simply write down formulas for the diffeomorphisms, and that's what is done in the lines I$\iff$II etc. The fact that I, II, III are not isometric to each other (meaning that the maps to not preserve the metrics) is irrelevant to the question of whether they are diffeomorphic to each other.
And you are right, the sphere picture needs that extra factor.